3.6.60 \(\int \frac {(d+e x)^{3/2}}{(a-c x^2)^3} \, dx\)

Optimal. Leaf size=268 \[ -\frac {3 \left (-2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 \left (2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\sqrt {d+e x} (a e-6 c d x)}{16 a^2 c \left (a-c x^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

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Rubi [A]  time = 0.51, antiderivative size = 268, 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 = {739, 823, 827, 1166, 208} \begin {gather*} -\frac {3 \left (-2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 \left (2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\sqrt {d+e x} (a e-6 c d x)}{16 a^2 c \left (a-c x^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a - c*x^2)^3,x]

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) - ((a*e - 6*c*d*x)*Sqrt[d + e*x])/(16*a^2*c*(a - c*x^2)) -
(3*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(3
2*a^(5/2)*c^(5/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + (3*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)
*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])

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 739

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*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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 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 {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (-6 c d^2+a e^2\right )-\frac {5}{2} c d e x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}+\frac {\int \frac {\frac {3}{4} c \left (c d^2-a e^2\right ) \left (4 c d^2-a e^2\right )+\frac {3}{2} c^2 d e \left (c d^2-a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2 \left (c d^2-a e^2\right )}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{2} c^2 d^2 e \left (c d^2-a e^2\right )+\frac {3}{4} c e \left (c d^2-a e^2\right ) \left (4 c d^2-a e^2\right )+\frac {3}{2} c^2 d e \left (c d^2-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 )}{4 a^2 c^2 \left (c d^2-a e^2\right )}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}-\frac {\left (3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} \sqrt {c}}+\frac {\left (3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} \sqrt {c}}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d+\sqrt {a} e}}\\ \end {align*}

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Mathematica [A]  time = 0.95, size = 428, normalized size = 1.60 \begin {gather*} \frac {\frac {2 (d+e x)^{5/2} \left (3 a^2 e^3-a c d e (5 d+4 e x)+6 c^2 d^3 x\right )}{a-c x^2}+\frac {3 \sqrt {\sqrt {a} e+\sqrt {c} d} \left (-a^{3/2} e^3+6 \sqrt {a} c d^2 e+a \sqrt {c} d e^2+4 c^{3/2} d^3\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )-3 \left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (a^{3/2} e^3-6 \sqrt {a} c d^2 e+a \sqrt {c} d e^2+4 c^{3/2} d^3\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )+2 \sqrt {a} \sqrt [4]{c} e \sqrt {d+e x} \left (3 a^2 e^4-a c d e^2 (13 d+4 e x)+6 c^2 d^3 (2 d+e x)\right )}{\sqrt {a} c^{5/4}}+\frac {8 a (d+e x)^{5/2} \left (c d^2-a e^2\right ) (c d x-a e)}{\left (a-c x^2\right )^2}}{32 a^2 \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(a - c*x^2)^3,x]

[Out]

((8*a*(c*d^2 - a*e^2)*(-(a*e) + c*d*x)*(d + e*x)^(5/2))/(a - c*x^2)^2 + (2*(d + e*x)^(5/2)*(3*a^2*e^3 + 6*c^2*
d^3*x - a*c*d*e*(5*d + 4*e*x)))/(a - c*x^2) + (2*Sqrt[a]*c^(1/4)*e*Sqrt[d + e*x]*(3*a^2*e^4 + 6*c^2*d^3*(2*d +
e*x) - a*c*d*e^2*(13*d + 4*e*x)) - 3*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(Sqrt[c]*d + Sqrt[a]*e)^2*(4*c^(3/2)*d^3 - 6
*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]
+ 3*(Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(4*c^(3/2)*d^3 + 6*Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^
2 - a^(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(5/4)))/(32*a^2*(c*d
^2 - a*e^2)^2)

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IntegrateAlgebraic [A]  time = 1.87, size = 382, normalized size = 1.43 \begin {gather*} \frac {3 \left (2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{32 a^{5/2} c \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {3 \left (-2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{32 a^{5/2} c \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {\sqrt {d+e x} \left (3 a^2 e^5-9 a c d^2 e^3+a c e^3 (d+e x)^2+8 a c d e^3 (d+e x)+6 c^2 d^4 e-18 c^2 d^3 e (d+e x)+18 c^2 d^2 e (d+e x)^2-6 c^2 d e (d+e x)^3\right )}{16 a^2 c \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(3/2)/(a - c*x^2)^3,x]

[Out]

(Sqrt[d + e*x]*(6*c^2*d^4*e - 9*a*c*d^2*e^3 + 3*a^2*e^5 - 18*c^2*d^3*e*(d + e*x) + 8*a*c*d*e^3*(d + e*x) + 18*
c^2*d^2*e*(d + e*x)^2 + a*c*e^3*(d + e*x)^2 - 6*c^2*d*e*(d + e*x)^3))/(16*a^2*c*(-(c*d^2) + a*e^2 + 2*c*d*(d +
e*x) - c*(d + e*x)^2)^2) + (3*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]
*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(32*a^(5/2)*c*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) - (3*(4*c
*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqr
t[a]*e)])/(32*a^(5/2)*c*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B]  time = 0.47, size = 1753, normalized size = 6.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(-c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 -
a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*
(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 - a^4*c*e^8 - (4*a^5*c^6*d^5
- 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c
^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2
+ a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 2
0*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5
*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*
a^3*c^2*d^2*e^6 - a^4*c*e^8 - (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(e^10/(a^5*c^7*d^4 - 2
*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2
)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))) + 3*(a^2*c^3*x^4 -
2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^1
0/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 - 12*a
*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 +
3*a^7*c^4*d*e^4)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^
2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5
*c^3*d^2 - a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*
d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 -
a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*a^3*c^2*d^2*e^6 - a^4*
c*e^8 + (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7
*c^5*e^4)))*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d
^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))) - 4*(6*c^2*d*x^3 - a*c*e*x^2 - 10*a*c*d*
x - 3*a^2*e)*sqrt(e*x + d))/(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)

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giac [B]  time = 0.62, size = 509, normalized size = 1.90 \begin {gather*} -\frac {3 \, {\left (4 \, \sqrt {a c} c^{3} d^{3} - 3 \, \sqrt {a c} a c^{2} d e^{2} - {\left (2 \, a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} d - \sqrt {a c} a^{3} c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e}} + \frac {3 \, {\left (4 \, \sqrt {a c} c^{3} d^{3} - 3 \, \sqrt {a c} a c^{2} d e^{2} + {\left (2 \, a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} d + \sqrt {a c} a^{3} c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{2} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - 6 \, \sqrt {x e + d} c^{2} d^{4} e - {\left (x e + d\right )}^{\frac {5}{2}} a c e^{3} - 8 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d e^{3} + 9 \, \sqrt {x e + d} a c d^{2} e^{3} - 3 \, \sqrt {x e + d} a^{2} e^{5}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(-c*x^2+a)^3,x, algorithm="giac")

[Out]

-3/32*(4*sqrt(a*c)*c^3*d^3 - 3*sqrt(a*c)*a*c^2*d*e^2 - (2*a*c^2*d^2*e - a^2*c*e^3)*abs(c))*arctan(sqrt(x*e + d
)/sqrt(-(a^2*c^2*d + sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^2*c^2))/(a^2*c^2)))/((a^3*c^3*d - sqrt(a*c
)*a^3*c^2*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)) + 3/32*(4*sqrt(a*c)*c^3*d^3 - 3*sqrt(a*c)*a*c^2*d*e^2 + (2*a*c^2*d^
2*e - a^2*c*e^3)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d - sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*
a^2*c^2))/(a^2*c^2)))/((a^3*c^3*d + sqrt(a*c)*a^3*c^2*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)) - 1/16*(6*(x*e + d)^(7/
2)*c^2*d*e - 18*(x*e + d)^(5/2)*c^2*d^2*e + 18*(x*e + d)^(3/2)*c^2*d^3*e - 6*sqrt(x*e + d)*c^2*d^4*e - (x*e +
d)^(5/2)*a*c*e^3 - 8*(x*e + d)^(3/2)*a*c*d*e^3 + 9*sqrt(x*e + d)*a*c*d^2*e^3 - 3*sqrt(x*e + d)*a^2*e^5)/(((x*e
+ d)^2*c - 2*(x*e + d)*c*d + c*d^2 - a*e^2)^2*a^2*c)

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maple [B]  time = 0.12, size = 608, normalized size = 2.27 \begin {gather*} -\frac {9 \sqrt {e x +d}\, d^{2} e^{3}}{16 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a}+\frac {3 \sqrt {e x +d}\, c \,d^{4} e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}+\frac {3 \sqrt {e x +d}\, e^{5}}{16 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} c}+\frac {\left (e x +d \right )^{\frac {3}{2}} d \,e^{3}}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a}-\frac {3 e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {3 e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {9 \left (e x +d \right )^{\frac {3}{2}} c \,d^{3} e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}+\frac {3 c \,d^{2} e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{8 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}+\frac {3 c \,d^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{8 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}+\frac {\left (e x +d \right )^{\frac {5}{2}} e^{3}}{16 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a}+\frac {9 \left (e x +d \right )^{\frac {5}{2}} c \,d^{2} e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}-\frac {3 \left (e x +d \right )^{\frac {7}{2}} c d e}{8 \left (c \,e^{2} x^{2}-a \,e^{2}\right )^{2} a^{2}}+\frac {3 d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{16 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}}-\frac {3 d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{16 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(-c*x^2+a)^3,x)

[Out]

-3/8*e/(c*e^2*x^2-a*e^2)^2*c*d/a^2*(e*x+d)^(7/2)+1/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)+9/8*e/(c*e^2*x^2
-a*e^2)^2/a^2*(e*x+d)^(5/2)*c*d^2+1/2*e^3/(c*e^2*x^2-a*e^2)^2*d/a*(e*x+d)^(3/2)-9/8*e/(c*e^2*x^2-a*e^2)^2*d^3/
a^2*(e*x+d)^(3/2)*c+3/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)-9/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*
d^2+3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(1/2)*d^4-3/32*e^3/a/(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)+3/8*e/a^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)*c*d^2+3/16*e/a^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-3/32*e^3/a/(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)+3/8*e/a^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)*c*d^2-3/16*e/a^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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(-c*x^2+a)^3,x, algorithm="maxima")

[Out]

-integrate((e*x + d)^(3/2)/(c*x^2 - a)^3, x)

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mupad [B]  time = 2.74, size = 3191, normalized size = 11.91

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(a - c*x^2)^3,x)

[Out]

(((4*a*d*e^3 - 9*c*d^3*e)*(d + e*x)^(3/2))/(8*a^2) + (e*(a*e^2 + 18*c*d^2)*(d + e*x)^(5/2))/(16*a^2) + (3*(d +
e*x)^(1/2)*(a^2*e^5 + 2*c^2*d^4*e - 3*a*c*d^2*e^3))/(16*a^2*c) - (3*c*d*e*(d + e*x)^(7/2))/(8*a^2))/(c^2*(d +
e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^
2*d*(d + e*x)^3 - 2*a*c*d^2*e^2) + atan(((((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4
*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4
096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*
a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^
2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^
3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i - (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048
*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*
c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7
*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6
+ 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4
+ 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i)/((3*(9*a^2*d*e^7 + 144*c^2*d^5*e^3 - 1
08*a*c*d^3*e^5))/(1024*a^6) + (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d +
e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c
^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3
*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^
2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(40
96*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c
^4*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/
(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 2
0*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*
e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*
d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^
3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*2i + atan(((((3*(2048*a^6*c^2*e^5 -
4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5
+ 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(a^15*c^5)^(1/
2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d
+ e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5
*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i - (((3*(2048*
a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 1
6*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(
a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2))
)^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*((9*(e^5*(a^15*c^5)^(
1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i)
/((3*(9*a^2*d*e^7 + 144*c^2*d^5*e^3 - 108*a*c*d^3*e^5))/(1024*a^6) + (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2
*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^
4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d
^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*
a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*
c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + (((3*(2048*a^6*c^2*e^5 - 4096*a
^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^
7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(a^15*c^5)^(1/2) + 16
*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) - ((d + e*x)
^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^
5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)))*((9*(e^5*(a^15*c^5)^(
1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*2i

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sympy [F(-1)]  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((e*x+d)**(3/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

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