3.6.37 \(\int \frac {1}{(d+e x)^{5/2} (a-c x^2)} \, dx\)

Optimal. Leaf size=190 \[ -\frac {c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}+\frac {4 c d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]

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Rubi [A]  time = 0.38, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {710, 829, 827, 1166, 208} \begin {gather*} -\frac {c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2}}+\frac {4 c d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

(2*e)/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (4*c*d*e)/((c*d^2 - a*e^2)^2*Sqrt[d + e*x]) - (c^(3/4)*ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)) + (c^(3/4)*ArcTanh[
(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^(5/2))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx &=\frac {2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {c \int \frac {d-e x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx}{c d^2-a e^2}\\ &=\frac {2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {c \int \frac {-c d^2-a e^2+2 c d e x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=\frac {2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {-2 c d^2 e+e \left (-c d^2-a e^2\right )+2 c d e x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (c d^2-a e^2\right )^2}\\ &=\frac {2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {c^{3/2} \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^2}+\frac {c^{3/2} \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^2}\\ &=\frac {2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {c^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 130, normalized size = 0.68 \begin {gather*} \frac {\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {c} d-\sqrt {a} e}-\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} e+\sqrt {c} d}}{3 \sqrt {a} (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

(Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[a]*e)]/(Sqrt[c]*d - Sqrt[a]*e) - Hyper
geometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]/(Sqrt[c]*d + Sqrt[a]*e))/(3*Sqrt[a]*(
d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.70, size = 254, normalized size = 1.34 \begin {gather*} \frac {2 \left (-a e^3+c d^2 e+6 c d e (d+e x)\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {c \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(5/2)*(a - c*x^2)),x]

[Out]

(2*(c*d^2*e - a*e^3 + 6*c*d*e*(d + e*x)))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (c*ArcTan[(Sqrt[-(c*d) - Sqr
t[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[-(Sqrt[c]*(Sq
rt[c]*d + Sqrt[a]*e))]) - (c*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])
/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B]  time = 0.57, size = 5142, normalized size = 27.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(-c*x^2+a),x, algorithm="fricas")

[Out]

1/6*(3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2
*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*
e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d
^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*
c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 12
0*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 +
10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 +
a^2*c^2*e^5)*sqrt(e*x + d) + (15*a*c^4*d^6*e^2 + 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (a*c^6*
d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 -
3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^
10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 2
52*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18
+ a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c
^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a
^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 -
120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*
e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^
6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) - 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d
^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^3*
d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*
c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^
4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12
*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d
^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2
*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2 + 35*a^
2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (a*c^6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4
*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4
+ 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^
16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*
c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^
2*c^2*d*e^4 + (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^
6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a
*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6
*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11
*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^1
0))) + 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e -
2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^
8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6
*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^
3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 -
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2
+ 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3
+ a^2*c^2*e^5)*sqrt(e*x + d) + (15*a*c^4*d^6*e^2 + 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (a*c^
6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10
- 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*
e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 -
252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^1
8 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3
*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110
*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4
- 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^
6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*
d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e^10))) - 3*(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2
*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt((c^4*d^5 + 10*a*c^
3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^
5*c*d^2*e^8 - a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 +
a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^
12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c
*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d
^2*e^8 - a^6*e^10))*log((5*c^4*d^4*e + 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2 + 35*
a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (a*c^6*d^13 - 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 + 20*a
^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 + 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e
^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*
d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^
8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt((c^4*d^5 + 10*a*c^3*d^3*e^2 + 5*
a^2*c^2*d*e^4 - (a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 -
a^6*e^10)*sqrt((25*c^7*d^8*e^2 + 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 + 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/
(a*c^10*d^20 - 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 - 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 - 252*a
^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 - 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 - 10*a^10*c*d^2*e^18 + a^
11*e^20)))/(a*c^5*d^10 - 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 - 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 - a^6*e
^10))) + 4*(6*c*d*e^2*x + 7*c*d^2*e - a*e^3)*sqrt(e*x + d))/(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*
e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)

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giac [B]  time = 0.53, size = 529, normalized size = 2.78 \begin {gather*} \frac {\sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} + \sqrt {{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{3 \, a c^{2} d^{2} e + \sqrt {a c} c^{2} d^{3} + 3 \, \sqrt {a c} a c d e^{2} + a^{2} c e^{3}} + \frac {\sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} - \sqrt {{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{3 \, a c^{2} d^{2} e - \sqrt {a c} c^{2} d^{3} - 3 \, \sqrt {a c} a c d e^{2} + a^{2} c e^{3}} + \frac {2 \, {\left (6 \, {\left (x e + d\right )} c d e + c d^{2} e - a e^{3}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(-c*x^2+a),x, algorithm="giac")

[Out]

sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 + sqrt
((c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(c^3*d^
4 - 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(c^3*d^4 - 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(3*a*c^2*d^2*e + sqrt(a*c)*c^2*d^
3 + 3*sqrt(a*c)*a*c*d*e^2 + a^2*c*e^3) + sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(c^3*d
^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 - sqrt((c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^6 - 3*a*c^2*d^4
*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(c^3*d^4 - 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(c^3*d^4 - 2*a*c^2*d^2*e^2 + a^2*c
*e^4)))/(3*a*c^2*d^2*e - sqrt(a*c)*c^2*d^3 - 3*sqrt(a*c)*a*c*d*e^2 + a^2*c*e^3) + 2/3*(6*(x*e + d)*c*d*e + c*d
^2*e - a*e^3)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(x*e + d)^(3/2))

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maple [B]  time = 0.07, size = 472, normalized size = 2.48 \begin {gather*} \frac {a \,c^{2} e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {a \,c^{2} e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {c^{3} d^{2} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {c^{3} d^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 c^{2} d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {2 c^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {4 c d e}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {2 e}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(-c*x^2+a),x)

[Out]

c^2/(a*e^2-c*d^2)^2/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2
))*c)^(1/2)*c)*a*e^3+e*c^3/(a*e^2-c*d^2)^2/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/
2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^2-2*e*c^2/(a*e^2-c*d^2)^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x
+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d+c^2/(a*e^2-c*d^2)^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^
(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*a*e^3+e*c^3/(a*e^2-c*d^2)^2/(a*c*e^2)^(1/2)/((-
c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d^2+2*e*c^2/(a*e^2-c*d^
2)^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d-2/3*e/(a*e^2-
c*d^2)/(e*x+d)^(3/2)+4*e*c*d/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {1}{{\left (c x^{2} - a\right )} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(-c*x^2+a),x, algorithm="maxima")

[Out]

-integrate(1/((c*x^2 - a)*(e*x + d)^(5/2)), x)

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mupad [B]  time = 2.66, size = 7831, normalized size = 41.22

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - c*x^2)*(d + e*x)^(5/2)),x)

[Out]

- ((2*e)/(3*(a*e^2 - c*d^2)) - (4*c*d*e*(d + e*x))/(a*e^2 - c*d^2)^2)/(d + e*x)^(3/2) - atan((((d + e*x)^(1/2)
*(16*a^8*c^5*e^18 + 16*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 1024*a^3*c^10*d^10*e^8 - 1440*a^4*c^9*d^8*e^10
+ 1024*a^5*c^8*d^6*e^12 - 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 1
0*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10
- 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(32*a^10*c^4*e^21 - (
d + e*x)^(1/2)*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^
3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e
^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 - 640*a^2*c^13*
d^19*e^4 + 2880*a^3*c^12*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 - 16128*a^6*c^9*d^11*e^1
2 + 13440*a^7*c^8*d^9*e^14 - 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 - 640*a^10*c^5*d^3*e^20) + 96*a*c^1
3*d^18*e^3 - 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11
- 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(a^3
*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*
(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a
^5*c^2*d^4*e^6)))^(1/2)*1i + ((d + e*x)^(1/2)*(16*a^8*c^5*e^18 + 16*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 10
24*a^3*c^10*d^10*e^8 - 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 - 320*a^6*c^7*d^4*e^14) - (-(a^2*e^5*(a^3
*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*
(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a
^5*c^2*d^4*e^6)))^(1/2)*(32*a^10*c^4*e^21 + (d + e*x)^(1/2)*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2
*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2
*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*
d^21*e^2 + 64*a^11*c^4*d*e^22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 1344
0*a^5*c^10*d^13*e^10 - 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 - 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6
*d^5*e^18 - 640*a^10*c^5*d^3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 448
0*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*
e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 +
5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 +
5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1i)/(((d + e*x)^(1/2)*(16*a^8*c^5*e^18 +
16*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 1024*a^3*c^10*d^10*e^8 - 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e
^12 - 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 +
5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 +
5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(32*a^10*c^4*e^21 - (d + e*x)^(1/2)*(-(a^
2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*
c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*
e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*
c^12*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 - 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^
9*e^14 - 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 - 640*a^10*c^5*d^3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2
*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e
^13 + 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*
d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*
(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1
/2) - ((d + e*x)^(1/2)*(16*a^8*c^5*e^18 + 16*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 1024*a^3*c^10*d^10*e^8 -
1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 - 320*a^6*c^7*d^4*e^14) - (-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5
+ 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^
7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)
*(32*a^10*c^4*e^21 + (d + e*x)^(1/2)*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3
*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2
*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*
d*e^22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 -
16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 - 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 - 640*a^10*c^
5*d^3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 4480*a^4*c^10*d^12*e^9 + 4
928*a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*
e^19))*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(
1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*
a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2) + 32*c^12*d^13*e^3 - 192*a*c^11*d^11*e^5 + 32*a^6*c^6*d*e^15 + 4
80*a^2*c^10*d^9*e^7 - 640*a^3*c^9*d^7*e^9 + 480*a^4*c^8*d^5*e^11 - 192*a^5*c^7*d^3*e^13))*(-(a^2*e^5*(a^3*c^3)
^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*
c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^
2*d^4*e^6)))^(1/2)*2i - atan((((d + e*x)^(1/2)*(16*a^8*c^5*e^18 + 16*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 1
024*a^3*c^10*d^10*e^8 - 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 - 320*a^6*c^7*d^4*e^14) + ((a^2*e^5*(a^3
*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*
(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a
^5*c^2*d^4*e^6)))^(1/2)*(32*a^10*c^4*e^21 - (d + e*x)^(1/2)*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*
d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*
c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d
^21*e^2 + 64*a^11*c^4*d*e^22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 13440
*a^5*c^10*d^13*e^10 - 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 - 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*
d^5*e^18 - 640*a^10*c^5*d^3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 4480
*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*e
^17 - 160*a^9*c^5*d^2*e^19))*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*
c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*
a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1i + ((d + e*x)^(1/2)*(16*a^8*c^5*e^18 + 16
*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 1024*a^3*c^10*d^10*e^8 - 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^1
2 - 320*a^6*c^7*d^4*e^14) - ((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c
^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a
^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(32*a^10*c^4*e^21 + (d + e*x)^(1/2)*((a^2*e^
5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^
2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4
+ 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12
*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 - 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^
14 - 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 - 640*a^10*c^5*d^3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2*c^1
2*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e^13
+ 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 -
5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*
e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1
i)/(((d + e*x)^(1/2)*(16*a^8*c^5*e^18 + 16*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 1024*a^3*c^10*d^10*e^8 - 14
40*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 - 320*a^6*c^7*d^4*e^14) + ((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 -
5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e
^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(3
2*a^10*c^4*e^21 - (d + e*x)^(1/2)*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2
+ 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8
+ 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^
22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 - 1612
8*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 - 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 - 640*a^10*c^5*d^
3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 4480*a^4*c^10*d^12*e^9 + 4928*
a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19
))*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2)
+ 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c
^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2) - ((d + e*x)^(1/2)*(16*a^8*c^5*e^18 + 16*c^13*d^16*e^2 - 320*a^2*c^11
*d^12*e^6 + 1024*a^3*c^10*d^10*e^8 - 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 - 320*a^6*c^7*d^4*e^14) - (
(a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10
*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d
^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(32*a^10*c^4*e^21 + (d + e*x)^(1/2)*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5
- 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^
7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)
*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 - 7680*a^4*c^11*d^1
5*e^8 + 13440*a^5*c^10*d^13*e^10 - 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 - 7680*a^8*c^7*d^7*e^16 +
2880*a^9*c^6*d^5*e^18 - 640*a^10*c^5*d^3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^
14*e^7 - 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 + 128*
a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3
*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c
*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2) + 32*c^12*d^13*e^3 - 192*a*c^1
1*d^11*e^5 + 32*a^6*c^6*d*e^15 + 480*a^2*c^10*d^9*e^7 - 640*a^3*c^9*d^7*e^9 + 480*a^4*c^8*d^5*e^11 - 192*a^5*c
^7*d^3*e^13))*((a^2*e^5*(a^3*c^3)^(1/2) - a*c^4*d^5 - 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*
c^3)^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2
- 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*2i

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- a d^{2} \sqrt {d + e x} - 2 a d e x \sqrt {d + e x} - a e^{2} x^{2} \sqrt {d + e x} + c d^{2} x^{2} \sqrt {d + e x} + 2 c d e x^{3} \sqrt {d + e x} + c e^{2} x^{4} \sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(-c*x**2+a),x)

[Out]

-Integral(1/(-a*d**2*sqrt(d + e*x) - 2*a*d*e*x*sqrt(d + e*x) - a*e**2*x**2*sqrt(d + e*x) + c*d**2*x**2*sqrt(d
+ e*x) + 2*c*d*e*x**3*sqrt(d + e*x) + c*e**2*x**4*sqrt(d + e*x)), x)

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