3.658 \(\int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=491 \[ -\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{32 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{32 c^2 d \sqrt{a+c x^2} \left (2 a e^2+c d^2\right )}{35 e^3 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a e^2+7 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]

[Out]

(32*c^2*d*(c*d^2 + 2*a*e^2)*Sqrt[a + c*x^2])/(35*e^3*(c*d^2 + a*e^2)^2*Sqrt[d +
e*x]) - (4*c*(2*d*(2*c*d^2 + a*e^2) + e*(7*c*d^2 + 5*a*e^2)*x)*Sqrt[a + c*x^2])/
(35*e^3*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) - (2*(a + c*x^2)^(3/2))/(7*e*(d + e*x)^
(7/2)) + (32*Sqrt[-a]*c^(5/2)*d*(c*d^2 + 2*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)
/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]
*Sqrt[c]*d - a*e)])/(35*e^4*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*
d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*c^(3/2)*(4*c*d^2 + 5*a*e^2)*Sqrt
[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[Arc
Sin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)
])/(35*e^4*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.42552, antiderivative size = 491, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{32 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{32 c^2 d \sqrt{a+c x^2} \left (2 a e^2+c d^2\right )}{35 e^3 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a e^2+7 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(32*c^2*d*(c*d^2 + 2*a*e^2)*Sqrt[a + c*x^2])/(35*e^3*(c*d^2 + a*e^2)^2*Sqrt[d +
e*x]) - (4*c*(2*d*(2*c*d^2 + a*e^2) + e*(7*c*d^2 + 5*a*e^2)*x)*Sqrt[a + c*x^2])/
(35*e^3*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) - (2*(a + c*x^2)^(3/2))/(7*e*(d + e*x)^
(7/2)) + (32*Sqrt[-a]*c^(5/2)*d*(c*d^2 + 2*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)
/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]
*Sqrt[c]*d - a*e)])/(35*e^4*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*
d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*c^(3/2)*(4*c*d^2 + 5*a*e^2)*Sqrt
[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[Arc
Sin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)
])/(35*e^4*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.41561, size = 659, normalized size = 1.34 \[ \frac{2 \left (-e^2 \left (a+c x^2\right ) \left (-16 c^2 d (d+e x)^3 \left (2 a e^2+c d^2\right )-16 c d (d+e x) \left (a e^2+c d^2\right )^2+c (d+e x)^2 \left (15 a e^2+19 c d^2\right ) \left (a e^2+c d^2\right )+5 \left (a e^2+c d^2\right )^3\right )-\frac{4 c^2 (d+e x)^3 \left (-\sqrt{a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 \sqrt{c} d (d+e x)^{3/2} \left (2 a^{3/2} e^3+\sqrt{a} c d^2 e-2 i a \sqrt{c} d e^2-i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (2 a^2 e^2+a c \left (d^2+2 e^2 x^2\right )+c^2 d^2 x^2\right )\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{35 e^5 \sqrt{a+c x^2} (d+e x)^{7/2} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(2*(-(e^2*(a + c*x^2)*(5*(c*d^2 + a*e^2)^3 - 16*c*d*(c*d^2 + a*e^2)^2*(d + e*x)
+ c*(c*d^2 + a*e^2)*(19*c*d^2 + 15*a*e^2)*(d + e*x)^2 - 16*c^2*d*(c*d^2 + 2*a*e^
2)*(d + e*x)^3)) - (4*c^2*(d + e*x)^3*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*
(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + 2*e^2*x^2)) + 4*Sqrt[c]*d*((-I)*c^(3/2)*d^
3 + Sqrt[a]*c*d^2*e - (2*I)*a*Sqrt[c]*d*e^2 + 2*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a]
)/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d +
e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]],
(Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] - Sqrt[a]*e*(4*c^(3/2)*d^3
+ I*Sqrt[a]*c*d^2*e + 8*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt[a
])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d
+ e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]]
, (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)
/Sqrt[c]]))/(35*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2)*Sqrt[a + c*x^2])

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Maple [B]  time = 0.092, size = 5277, normalized size = 10.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d)^(9/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(3/2)/(e*x + d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^(3/2)/((e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x +
d^4)*sqrt(e*x + d)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(3/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError