3.7.30 \(\int \frac {1}{(d+e x)^{5/2} (a-c x^2)^2} \, dx\) [630]

Optimal. Leaf size=311 \[ -\frac {e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}-\frac {c^{3/4} \left (2 \sqrt {c} d-7 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2}}+\frac {c^{3/4} \left (2 \sqrt {c} d+7 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d+\sqrt {a} e\right )^{7/2}} \]

[Out]

-1/6*e*(7*a*e^2+3*c*d^2)/a/(-a*e^2+c*d^2)^2/(e*x+d)^(3/2)+1/2*(c*d*x-a*e)/a/(-a*e^2+c*d^2)/(e*x+d)^(3/2)/(-c*x
^2+a)-1/4*c^(3/4)*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(-7*e*a^(1/2)+2*d*c^(1/2))/a^(3/
2)/(-e*a^(1/2)+d*c^(1/2))^(7/2)+1/4*c^(3/4)*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(7*e*a^
(1/2)+2*d*c^(1/2))/a^(3/2)/(e*a^(1/2)+d*c^(1/2))^(7/2)-1/2*c*d*e*(19*a*e^2+c*d^2)/a/(-a*e^2+c*d^2)^3/(e*x+d)^(
1/2)

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Rubi [A]
time = 0.75, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {755, 843, 841, 1180, 214} \begin {gather*} -\frac {c^{3/4} \left (2 \sqrt {c} d-7 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2}}+\frac {c^{3/4} \left (7 \sqrt {a} e+2 \sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )^{7/2}}-\frac {a e-c d x}{2 a \left (a-c x^2\right ) (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {e \left (7 a e^2+3 c d^2\right )}{6 a (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac {c d e \left (19 a e^2+c d^2\right )}{2 a \sqrt {d+e x} \left (c d^2-a e^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(e*(3*c*d^2 + 7*a*e^2))/(a*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*d*e*(c*d^2 + 19*a*e^2))/(2*a*(c*d^2 -
a*e^2)^3*Sqrt[d + e*x]) - (a*e - c*d*x)/(2*a*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a - c*x^2)) - (c^(3/4)*(2*Sqrt[c
]*d - 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*(Sqrt[c]*d - Sqrt[
a]*e)^(7/2)) + (c^(3/4)*(2*Sqrt[c]*d + 7*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e
]])/(4*a^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)^(7/2))

Rule 214

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

Rule 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 841

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 843

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 1180

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 )^2} \, dx &=-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{2} \left (2 c d^2-7 a e^2\right )+\frac {5}{2} c d e x}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )}\\ &=-\frac {e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}-\frac {\int \frac {-c d \left (c d^2-6 a e^2\right )-\frac {1}{2} c e \left (3 c d^2+7 a e^2\right ) x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )^2}\\ &=-\frac {e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{2} c \left (2 c^2 d^4-15 a c d^2 e^2-7 a^2 e^4\right )+\frac {1}{2} c^2 d e \left (c d^2+19 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )^3}\\ &=-\frac {e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} c^2 d^2 e \left (c d^2+19 a e^2\right )+\frac {1}{2} c e \left (2 c^2 d^4-15 a c d^2 e^2-7 a^2 e^4\right )+\frac {1}{2} c^2 d e \left (c d^2+19 a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a \left (c d^2-a e^2\right )^3}\\ &=-\frac {e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}-\frac {\left (c^{3/2} \left (2 \sqrt {c} d-7 \sqrt {a} e\right )\right ) \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^3}+\frac {\left (c^{3/2} \left (2 \sqrt {c} d+7 \sqrt {a} e\right )\right ) \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} \left (\sqrt {c} d+\sqrt {a} e\right )^3}\\ &=-\frac {e \left (3 c d^2+7 a e^2\right )}{6 a \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d e \left (c d^2+19 a e^2\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {a e-c d x}{2 a \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a-c x^2\right )}-\frac {c^{3/4} \left (2 \sqrt {c} d-7 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2}}+\frac {c^{3/4} \left (2 \sqrt {c} d+7 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} \left (\sqrt {c} d+\sqrt {a} e\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.17, size = 359, normalized size = 1.15 \begin {gather*} \frac {-\frac {2 \sqrt {a} \left (4 a^3 e^5+3 c^3 d^3 x (d+e x)^2-a^2 c e^3 \left (55 d^2+54 d e x+7 e^2 x^2\right )+a c^2 d e \left (-9 d^3-9 d^2 e x+61 d e^2 x^2+57 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (-a+c x^2\right )}+\frac {3 c \left (2 \sqrt {c} d+7 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {3 c \left (2 \sqrt {c} d-7 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right )^3 \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{12 a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[a]*(4*a^3*e^5 + 3*c^3*d^3*x*(d + e*x)^2 - a^2*c*e^3*(55*d^2 + 54*d*e*x + 7*e^2*x^2) + a*c^2*d*e*(-9*
d^3 - 9*d^2*e*x + 61*d*e^2*x^2 + 57*e^3*x^3)))/((c*d^2 - a*e^2)^3*(d + e*x)^(3/2)*(-a + c*x^2)) + (3*c*(2*Sqrt
[c]*d + 7*Sqrt[a]*e)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/((Sqrt[
c]*d + Sqrt[a]*e)^3*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]) - (3*c*(2*Sqrt[c]*d - 7*Sqrt[a]*e)*ArcTan[(Sqrt[-(c*d) +
Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/((Sqrt[c]*d - Sqrt[a]*e)^3*Sqrt[-(c*d) + Sqrt[a]*
Sqrt[c]*e]))/(12*a^(3/2))

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Maple [A]
time = 0.45, size = 402, normalized size = 1.29

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {c \left (\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}+\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-7 a^{2} e^{4}-15 a c \,d^{2} e^{2}+2 c^{2} d^{4}+19 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (7 a^{2} e^{4}+15 a c \,d^{2} e^{2}-2 c^{2} d^{4}+19 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 d c}{\left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {e x +d}}\right )\)	\(402\)
default	\(2 e^{3} \left (\frac {c \left (\frac {-\frac {c d \left (3 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 a \,e^{2}}+\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}}{4 a \,e^{2}}}{-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {c \left (-\frac {\left (-7 a^{2} e^{4}-15 a c \,d^{2} e^{2}+2 c^{2} d^{4}+19 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (7 a^{2} e^{4}+15 a c \,d^{2} e^{2}-2 c^{2} d^{4}+19 \sqrt {a c \,e^{2}}\, a d \,e^{2}+\sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 a \,e^{2}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 d c}{\left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {e x +d}}\right )\)	\(402\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3*(c/(a*e^2-c*d^2)^3*((-1/4*c*d*(3*a*e^2+c*d^2)/a/e^2*(e*x+d)^(3/2)+1/4*(a^2*e^4+6*a*c*d^2*e^2+c^2*d^4)/a/
e^2*(e*x+d)^(1/2))/(-c*(e*x+d)^2+2*c*d*(e*x+d)+e^2*a-c*d^2)+1/4/a/e^2*c*(-1/2*(-7*a^2*e^4-15*a*c*d^2*e^2+2*c^2
*d^4+19*(a*c*e^2)^(1/2)*a*d*e^2+(a*c*e^2)^(1/2)*c*d^3)/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh
(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+1/2*(7*a^2*e^4+15*a*c*d^2*e^2-2*c^2*d^4+19*(a*c*e^2)^(1/2)*a
*d*e^2+(a*c*e^2)^(1/2)*c*d^3)/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(
a*c*e^2)^(1/2))*c)^(1/2))))-1/3/(a*e^2-c*d^2)^2/(e*x+d)^(3/2)+4/(a*e^2-c*d^2)^3*d*c/(e*x+d)^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7848 vs. \(2 (257) = 514\).
time = 10.85, size = 7848, normalized size = 25.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(a*c^4*d^8*x^2 - a^2*c^3*d^8 - (a^4*c*x^4 - a^5*x^2)*e^8 - 2*(a^4*c*d*x^3 - a^5*d*x)*e^7 + (3*a^3*c^2*
d^2*x^4 - 4*a^4*c*d^2*x^2 + a^5*d^2)*e^6 + 6*(a^3*c^2*d^3*x^3 - a^4*c*d^3*x)*e^5 - 3*(a^2*c^3*d^4*x^4 - 2*a^3*
c^2*d^4*x^2 + a^4*c*d^4)*e^4 - 6*(a^2*c^3*d^5*x^3 - a^3*c^2*d^5*x)*e^3 + (a*c^4*d^6*x^4 - 4*a^2*c^3*d^6*x^2 +
3*a^3*c^2*d^6)*e^2 + 2*(a*c^4*d^7*x^3 - a^2*c^3*d^7*x)*e)*sqrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5
*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 3
5*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d
^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14
+ 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 -
364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c
^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e
^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5
*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))*log
((420*c^6*d^8*e^3 - 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*e^7 + 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sq
rt(x*e + d) + (105*a^2*c^6*d^10*e^4 - 4389*a^3*c^5*d^8*e^6 + 26274*a^4*c^4*d^6*e^8 + 34142*a^5*c^3*d^4*e^10 +
7525*a^6*c^2*d^2*e^12 + 343*a^7*c*e^14 + 2*(a^3*c^9*d^19 - 15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 - 112*a^6
*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 + 154*a^8*c^4*d^9*e^10 - 280*a^9*c^3*d^7*e^12 + 216*a^10*c^2*d^5*e^14 - 83
*a^11*c*d^3*e^16 + 13*a^12*d*e^18)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10
+ 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d
^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*
c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^1
0*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^2
8)))*sqrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^
3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d
^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*
e^10 + 1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c
^14*d^28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002
*a^8*c^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^
5*d^10*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^1
7*e^28)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 -
21*a^8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))) - 3*(a*c^4*d^8*x^2 - a^2*c^3*d^8 - (a^4*c*x^4 - a^5*x^2
)*e^8 - 2*(a^4*c*d*x^3 - a^5*d*x)*e^7 + (3*a^3*c^2*d^2*x^4 - 4*a^4*c*d^2*x^2 + a^5*d^2)*e^6 + 6*(a^3*c^2*d^3*x
^3 - a^4*c*d^3*x)*e^5 - 3*(a^2*c^3*d^4*x^4 - 2*a^3*c^2*d^4*x^2 + a^4*c*d^4)*e^4 - 6*(a^2*c^3*d^5*x^3 - a^3*c^2
*d^5*x)*e^3 + (a*c^4*d^6*x^4 - 4*a^2*c^3*d^6*x^2 + 3*a^3*c^2*d^6)*e^2 + 2*(a*c^4*d^7*x^3 - a^2*c^3*d^7*x)*e)*s
qrt((4*c^6*d^9 - 63*a*c^5*d^7*e^2 + 189*a^2*c^4*d^5*e^4 + 1155*a^3*c^3*d^3*e^6 + 315*a^4*c^2*d*e^8 + (a^3*c^7*
d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^8*c^2*d^4*e^1
0 + 7*a^9*c*d^2*e^12 - a^10*e^14)*sqrt((11025*c^9*d^12*e^6 - 171990*a*c^8*d^10*e^8 + 494991*a^2*c^7*d^8*e^10 +
1360716*a^3*c^6*d^6*e^12 + 780831*a^4*c^5*d^4*e^14 + 82026*a^5*c^4*d^2*e^16 + 2401*a^6*c^3*e^18)/(a^3*c^14*d^
28 - 14*a^4*c^13*d^26*e^2 + 91*a^5*c^12*d^24*e^4 - 364*a^6*c^11*d^22*e^6 + 1001*a^7*c^10*d^20*e^8 - 2002*a^8*c
^9*d^18*e^10 + 3003*a^9*c^8*d^16*e^12 - 3432*a^10*c^7*d^14*e^14 + 3003*a^11*c^6*d^12*e^16 - 2002*a^12*c^5*d^10
*e^18 + 1001*a^13*c^4*d^8*e^20 - 364*a^14*c^3*d^6*e^22 + 91*a^15*c^2*d^4*e^24 - 14*a^16*c*d^2*e^26 + a^17*e^28
)))/(a^3*c^7*d^14 - 7*a^4*c^6*d^12*e^2 + 21*a^5*c^5*d^10*e^4 - 35*a^6*c^4*d^8*e^6 + 35*a^7*c^3*d^6*e^8 - 21*a^
8*c^2*d^4*e^10 + 7*a^9*c*d^2*e^12 - a^10*e^14))*log((420*c^6*d^8*e^3 - 8421*a*c^5*d^6*e^5 + 36783*a^2*c^4*d^4*
e^7 + 40817*a^3*c^3*d^2*e^9 + 2401*a^4*c^2*e^11)*sqrt(x*e + d) - (105*a^2*c^6*d^10*e^4 - 4389*a^3*c^5*d^8*e^6
+ 26274*a^4*c^4*d^6*e^8 + 34142*a^5*c^3*d^4*e^10 + 7525*a^6*c^2*d^2*e^12 + 343*a^7*c*e^14 + 2*(a^3*c^9*d^19 -
15*a^4*c^8*d^17*e^2 + 64*a^5*c^7*d^15*e^4 - 112*a^6*c^6*d^13*e^6 + 42*a^7*c^5*d^11*e^8 + 154*a^8*c^4*d^9*e^10
- 280*a^9*c^3*d^7*e^12 + 216*a^10*c^2*d^5*e^14 ...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1872 vs. \(2 (257) = 514\).
time = 3.36, size = 1872, normalized size = 6.02 \begin {gather*} \frac {{\left ({\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7}\right )}^{2} {\left (\sqrt {a c} c d^{3} e + 19 \, \sqrt {a c} a d e^{3}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c \right |} - {\left (a c^{5} d^{10} e - 37 \, a^{2} c^{4} d^{8} e^{3} + 98 \, a^{3} c^{3} d^{6} e^{5} - 82 \, a^{4} c^{2} d^{4} e^{7} + 13 \, a^{5} c d^{2} e^{9} + 7 \, a^{6} e^{11}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} \right |} {\left | c \right |} - {\left (2 \, \sqrt {a c} a c^{8} d^{17} e - 27 \, \sqrt {a c} a^{2} c^{7} d^{15} e^{3} + 113 \, \sqrt {a c} a^{3} c^{6} d^{13} e^{5} - 223 \, \sqrt {a c} a^{4} c^{5} d^{11} e^{7} + 225 \, \sqrt {a c} a^{5} c^{4} d^{9} e^{9} - 97 \, \sqrt {a c} a^{6} c^{3} d^{7} e^{11} - 13 \, \sqrt {a c} a^{7} c^{2} d^{5} e^{13} + 27 \, \sqrt {a c} a^{8} c d^{3} e^{15} - 7 \, \sqrt {a c} a^{9} d e^{17}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{4} d^{7} - 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} - a^{4} c d e^{6} - \sqrt {{\left (a c^{4} d^{7} - 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} - a^{4} c d e^{6}\right )}^{2} - {\left (a c^{4} d^{8} - 4 \, a^{2} c^{3} d^{6} e^{2} + 6 \, a^{3} c^{2} d^{4} e^{4} - 4 \, a^{4} c d^{2} e^{6} + a^{5} e^{8}\right )} {\left (a c^{4} d^{6} - 3 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4} - a^{4} c e^{6}\right )}}}{a c^{4} d^{6} - 3 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4} - a^{4} c e^{6}}}}\right )}{4 \, {\left (a^{2} c^{8} d^{14} - 7 \, a^{3} c^{7} d^{12} e^{2} + 21 \, a^{4} c^{6} d^{10} e^{4} - 35 \, a^{5} c^{5} d^{8} e^{6} + 35 \, a^{6} c^{4} d^{6} e^{8} - 21 \, a^{7} c^{3} d^{4} e^{10} + 7 \, a^{8} c^{2} d^{2} e^{12} - a^{9} c e^{14}\right )} {\left | a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} \right |}} - \frac {{\left ({\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7}\right )}^{2} {\left (c^{2} d^{3} e + 19 \, a c d e^{3}\right )} {\left | c \right |} + {\left (\sqrt {a c} c^{5} d^{10} e - 37 \, \sqrt {a c} a c^{4} d^{8} e^{3} + 98 \, \sqrt {a c} a^{2} c^{3} d^{6} e^{5} - 82 \, \sqrt {a c} a^{3} c^{2} d^{4} e^{7} + 13 \, \sqrt {a c} a^{4} c d^{2} e^{9} + 7 \, \sqrt {a c} a^{5} e^{11}\right )} {\left | a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} \right |} {\left | c \right |} - {\left (2 \, a c^{9} d^{17} e - 27 \, a^{2} c^{8} d^{15} e^{3} + 113 \, a^{3} c^{7} d^{13} e^{5} - 223 \, a^{4} c^{6} d^{11} e^{7} + 225 \, a^{5} c^{5} d^{9} e^{9} - 97 \, a^{6} c^{4} d^{7} e^{11} - 13 \, a^{7} c^{3} d^{5} e^{13} + 27 \, a^{8} c^{2} d^{3} e^{15} - 7 \, a^{9} c d e^{17}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{4} d^{7} - 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} - a^{4} c d e^{6} + \sqrt {{\left (a c^{4} d^{7} - 3 \, a^{2} c^{3} d^{5} e^{2} + 3 \, a^{3} c^{2} d^{3} e^{4} - a^{4} c d e^{6}\right )}^{2} - {\left (a c^{4} d^{8} - 4 \, a^{2} c^{3} d^{6} e^{2} + 6 \, a^{3} c^{2} d^{4} e^{4} - 4 \, a^{4} c d^{2} e^{6} + a^{5} e^{8}\right )} {\left (a c^{4} d^{6} - 3 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4} - a^{4} c e^{6}\right )}}}{a c^{4} d^{6} - 3 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4} - a^{4} c e^{6}}}}\right )}{4 \, {\left (a^{2} c^{6} d^{12} e - \sqrt {a c} a c^{6} d^{13} + 6 \, \sqrt {a c} a^{2} c^{5} d^{11} e^{2} - 6 \, a^{3} c^{5} d^{10} e^{3} - 15 \, \sqrt {a c} a^{3} c^{4} d^{9} e^{4} + 15 \, a^{4} c^{4} d^{8} e^{5} + 20 \, \sqrt {a c} a^{4} c^{3} d^{7} e^{6} - 20 \, a^{5} c^{3} d^{6} e^{7} - 15 \, \sqrt {a c} a^{5} c^{2} d^{5} e^{8} + 15 \, a^{6} c^{2} d^{4} e^{9} + 6 \, \sqrt {a c} a^{6} c d^{3} e^{10} - 6 \, a^{7} c d^{2} e^{11} - \sqrt {a c} a^{7} d e^{12} + a^{8} e^{13}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - \sqrt {x e + d} c^{3} d^{4} e + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d e^{3} - 6 \, \sqrt {x e + d} a c^{2} d^{2} e^{3} - \sqrt {x e + d} a^{2} c e^{5}}{2 \, {\left (a c^{3} d^{6} - 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} - a^{4} e^{6}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}} - \frac {2 \, {\left (12 \, {\left (x e + d\right )} c d e^{3} + c d^{2} e^{3} - a e^{5}\right )}}{3 \, {\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 (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)^2,x, algorithm="giac")

[Out]

1/4*((a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)^2*(sqrt(a*c)*c*d^3*e + 19*sqrt(a*c)*a*d*e^3
)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(c) - (a*c^5*d^10*e - 37*a^2*c^4*d^8*e^3 + 98*a^3*c^3*d^6*e^5 - 82*a^4*c^2*d
^4*e^7 + 13*a^5*c*d^2*e^9 + 7*a^6*e^11)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a
^3*c*d^2*e^5 - a^4*e^7)*abs(c) - (2*sqrt(a*c)*a*c^8*d^17*e - 27*sqrt(a*c)*a^2*c^7*d^15*e^3 + 113*sqrt(a*c)*a^3
*c^6*d^13*e^5 - 223*sqrt(a*c)*a^4*c^5*d^11*e^7 + 225*sqrt(a*c)*a^5*c^4*d^9*e^9 - 97*sqrt(a*c)*a^6*c^3*d^7*e^11
- 13*sqrt(a*c)*a^7*c^2*d^5*e^13 + 27*sqrt(a*c)*a^8*c*d^3*e^15 - 7*sqrt(a*c)*a^9*d*e^17)*sqrt(-c^2*d + sqrt(a*
c)*c*e)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6 -
sqrt((a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6)^2 - (a*c^4*d^8 - 4*a^2*c^3*d^6*e^2 + 6*
a^3*c^2*d^4*e^4 - 4*a^4*c*d^2*e^6 + a^5*e^8)*(a*c^4*d^6 - 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e^6)))
/(a*c^4*d^6 - 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e^6)))/((a^2*c^8*d^14 - 7*a^3*c^7*d^12*e^2 + 21*a^
4*c^6*d^10*e^4 - 35*a^5*c^5*d^8*e^6 + 35*a^6*c^4*d^6*e^8 - 21*a^7*c^3*d^4*e^10 + 7*a^8*c^2*d^2*e^12 - a^9*c*e^
14)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)) - 1/4*((a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3
+ 3*a^3*c*d^2*e^5 - a^4*e^7)^2*(c^2*d^3*e + 19*a*c*d*e^3)*abs(c) + (sqrt(a*c)*c^5*d^10*e - 37*sqrt(a*c)*a*c^4*
d^8*e^3 + 98*sqrt(a*c)*a^2*c^3*d^6*e^5 - 82*sqrt(a*c)*a^3*c^2*d^4*e^7 + 13*sqrt(a*c)*a^4*c*d^2*e^9 + 7*sqrt(a*
c)*a^5*e^11)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)*abs(c) - (2*a*c^9*d^17*e - 27*a^
2*c^8*d^15*e^3 + 113*a^3*c^7*d^13*e^5 - 223*a^4*c^6*d^11*e^7 + 225*a^5*c^5*d^9*e^9 - 97*a^6*c^4*d^7*e^11 - 13*
a^7*c^3*d^5*e^13 + 27*a^8*c^2*d^3*e^15 - 7*a^9*c*d*e^17)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^4*d^7 - 3*a^2
*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4*c*d*e^6 + sqrt((a*c^4*d^7 - 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 - a^4
*c*d*e^6)^2 - (a*c^4*d^8 - 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 - 4*a^4*c*d^2*e^6 + a^5*e^8)*(a*c^4*d^6 - 3*a
^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e^6)))/(a*c^4*d^6 - 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 - a^4*c*e
^6)))/((a^2*c^6*d^12*e - sqrt(a*c)*a*c^6*d^13 + 6*sqrt(a*c)*a^2*c^5*d^11*e^2 - 6*a^3*c^5*d^10*e^3 - 15*sqrt(a*
c)*a^3*c^4*d^9*e^4 + 15*a^4*c^4*d^8*e^5 + 20*sqrt(a*c)*a^4*c^3*d^7*e^6 - 20*a^5*c^3*d^6*e^7 - 15*sqrt(a*c)*a^5
*c^2*d^5*e^8 + 15*a^6*c^2*d^4*e^9 + 6*sqrt(a*c)*a^6*c*d^3*e^10 - 6*a^7*c*d^2*e^11 - sqrt(a*c)*a^7*d*e^12 + a^8
*e^13)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7)) - 1/2*((
x*e + d)^(3/2)*c^3*d^3*e - sqrt(x*e + d)*c^3*d^4*e + 3*(x*e + d)^(3/2)*a*c^2*d*e^3 - 6*sqrt(x*e + d)*a*c^2*d^2
*e^3 - sqrt(x*e + d)*a^2*c*e^5)/((a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*((x*e + d)^2*c -
2*(x*e + d)*c*d + c*d^2 - a*e^2)) - 2/3*(12*(x*e + d)*c*d*e^3 + c*d^2*e^3 - a*e^5)/((c^3*d^6 - 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 - a^3*e^6)*(x*e + d)^(3/2))

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Mupad [B]
time = 4.84, size = 2500, normalized size = 8.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*e^3)/(3*(a*e^2 - c*d^2)) - (20*c*d*e^3*(d + e*x))/(3*(a*e^2 - c*d^2)^2) - (c*e*(d + e*x)^2*(7*a^2*e^4 +
3*c^2*d^4 + 110*a*c*d^2*e^2))/(6*a*(a*e^2 - c*d^2)^3) + (c^2*d*e*(19*a*e^2 + c*d^2)*(d + e*x)^3)/(2*a*(a*e^2 -
c*d^2)^3))/((a*e^2 - c*d^2)*(d + e*x)^(3/2) - c*(d + e*x)^(7/2) + 2*c*d*(d + e*x)^(5/2)) - atan((((d + e*x)^(
1/2)*(1568*a^16*c^5*e^28 - 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 - 29120*a^5*c^16*d^22*e^6 + 128128*a
^6*c^15*d^20*e^8 - 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 + 282240*a^9*c^12*d^14*e^14 - 1059840
*a^10*c^11*d^12*e^16 + 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 + 456512*a^13*c^8*d^6*e^22 - 10
0480*a^14*c^7*d^4*e^24 + 4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2
*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 8
19*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c
*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*
d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*
c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5
*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^
7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2
)*(2048*a^21*c^4*d*e^32 - 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 - 215040*a^8*c^17*d^27*e^6 + 931840
*a^9*c^16*d^25*e^8 - 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 - 10250240*a^12*c^13*d^19*e^14
+ 13178880*a^13*c^12*d^17*e^16 - 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 - 6150144*a^16*c^
9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 - 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 - 30720*a^20*c^5
*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 - 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 -
609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 - 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*
e^15 - 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 - 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c
^8*d^8*e^23 - 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 + 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9
- 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3
*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2)
)/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^
8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i + ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 - 128*a^
3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 - 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 - 282560*a^7*c^1
4*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 + 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 + 1403904*a^
11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 + 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 + 4160*a^1
5*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189
*a^5*c^4*d^5*e^4 + 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2)
- 837*a^2*c*d^2*e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 -
21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2
)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4
+ 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(a^9*c^3)^(1/2) - 837*a^2*c*d^2*
e^7*(a^9*c^3)^(1/2))/(64*(a^13*e^14 - a^6*c^7*d^14 - 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 - 21*a^8*c^5*d^10*
e^4 + 35*a^9*c^4*d^8*e^6 - 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 - 2048*a^
6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 - 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 - 2795520*a^10
*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 - 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 -
13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 - 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^
9*e^24 - 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 - 30720*a^20*c^5*d^3*e^30) + 1792*a^19*c^4*e^31 -
256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 - 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 - 19233
28*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 - 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^1
7 - 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 - 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6
*e^25 + 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(a^9*c^3)^(1/2) + 31
5*a^7*c^2*d*e^8 - 63*a^4*c^5*d^7*e^2 + 189*a^5*...

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