3.664 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=498 \[ -\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (29 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (5 a e^2+8 c d^2\right )+d \left (29 a e^2+32 c d^2\right )\right )}{21 e^5 \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (5 a e^2+11 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]

[Out]

(8*c^2*(d*(32*c*d^2 + 29*a*e^2) + e*(8*c*d^2 + 5*a*e^2)*x)*Sqrt[a + c*x^2])/(21*
e^5*(c*d^2 + a*e^2)*Sqrt[d + e*x]) - (4*c*(2*d*(4*c*d^2 + a*e^2) + e*(11*c*d^2 +
5*a*e^2)*x)*(a + c*x^2)^(3/2))/(21*e^3*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) - (2*(a
+ c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2)) + (16*Sqrt[-a]*c^(5/2)*d*(32*c*d^2 + 29*a
*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sq
rt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(21*e^6*(c*d^2 + a*e^2)*
Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-
a]*c^(3/2)*(32*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[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]],
(-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(21*e^6*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.40331, antiderivative size = 498, 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 \[ -\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (29 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (5 a e^2+8 c d^2\right )+d \left (29 a e^2+32 c d^2\right )\right )}{21 e^5 \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (5 a e^2+11 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(8*c^2*(d*(32*c*d^2 + 29*a*e^2) + e*(8*c*d^2 + 5*a*e^2)*x)*Sqrt[a + c*x^2])/(21*
e^5*(c*d^2 + a*e^2)*Sqrt[d + e*x]) - (4*c*(2*d*(4*c*d^2 + a*e^2) + e*(11*c*d^2 +
5*a*e^2)*x)*(a + c*x^2)^(3/2))/(21*e^3*(c*d^2 + a*e^2)*(d + e*x)^(5/2)) - (2*(a
+ c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2)) + (16*Sqrt[-a]*c^(5/2)*d*(32*c*d^2 + 29*a
*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sq
rt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(21*e^6*(c*d^2 + a*e^2)*
Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-
a]*c^(3/2)*(32*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[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]],
(-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(21*e^6*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)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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Mathematica [C]  time = 8.27306, size = 677, normalized size = 1.36 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (\frac{2 c^2 d \left (67 a e^2+79 c d^2\right )}{(d+e x) \left (a e^2+c d^2\right )}-\frac{4 c \left (4 a e^2+13 c d^2\right )}{(d+e x)^2}+\frac{18 c d \left (a e^2+c d^2\right )}{(d+e x)^3}-\frac{3 \left (a e^2+c d^2\right )^2}{(d+e x)^4}+7 c^2\right )}{e^5}-\frac{16 c^2 \left (-\sqrt{a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+8 i \sqrt{a} c d^2 e+29 a \sqrt{c} d e^2+32 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+\sqrt{c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3+32 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2-32 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (29 a^2 e^2+a c \left (32 d^2+29 e^2 x^2\right )+32 c^2 d^2 x^2\right )\right )}{e^7 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (a e^2+c d^2\right )}\right )}{21 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(7*c^2 - (3*(c*d^2 + a*e^2)^2)/(d + e*x)^4 + (18*
c*d*(c*d^2 + a*e^2))/(d + e*x)^3 - (4*c*(13*c*d^2 + 4*a*e^2))/(d + e*x)^2 + (2*c
^2*d*(79*c*d^2 + 67*a*e^2))/((c*d^2 + a*e^2)*(d + e*x))))/e^5 - (16*c^2*(d*e^2*S
qrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(29*a^2*e^2 + 32*c^2*d^2*x^2 + a*c*(32*d^2 + 29*
e^2*x^2)) + Sqrt[c]*d*((-32*I)*c^(3/2)*d^3 + 32*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[
c]*d*e^2 + 29*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[Sq
rt[-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*(32*c^(3/2)*d^3 + (8*I)*Sqrt[a]*c*d^2*e + 29*a*S
qrt[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*Arc
Sinh[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)]))/(e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^
2)*(d + e*x))))/(21*Sqrt[a + c*x^2])

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Maple [B]  time = 0.056, size = 5303, normalized size = 10.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(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. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(9/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)/((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)**(5/2)/(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)^(5/2)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError