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

Optimal. Leaf size=217 \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]

[Out]

-(a^2*Sqrt[c + d*x^2])/(7*b^2*(b*c - a*d)*(a + b*x^2)^(7/2)) + (2*a*(7*b*c - 4*a
*d)*Sqrt[c + d*x^2])/(35*b^2*(b*c - a*d)^2*(a + b*x^2)^(5/2)) - ((35*b^2*c^2 - 1
4*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(105*b^2*(b*c - a*d)^3*(a + b*x^2)^(3/2)
) + (2*d*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(105*b^2*(b*c -
a*d)^4*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.648056, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*Sqrt[c + d*x^2])/(7*b^2*(b*c - a*d)*(a + b*x^2)^(7/2)) + (2*a*(7*b*c - 4*a
*d)*Sqrt[c + d*x^2])/(35*b^2*(b*c - a*d)^2*(a + b*x^2)^(5/2)) - ((35*b^2*c^2 - 1
4*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(105*b^2*(b*c - a*d)^3*(a + b*x^2)^(3/2)
) + (2*d*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(105*b^2*(b*c -
a*d)^4*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 62.3711, size = 204, normalized size = 0.94 \[ \frac{a^{2} \sqrt{c + d x^{2}}}{7 b^{2} \left (a + b x^{2}\right )^{\frac{7}{2}} \left (a d - b c\right )} - \frac{2 a \sqrt{c + d x^{2}} \left (4 a d - 7 b c\right )}{35 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}} + \frac{2 d \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} - 14 a b c d + 35 b^{2} c^{2}\right )}{105 b^{2} \sqrt{a + b x^{2}} \left (a d - b c\right )^{4}} + \frac{\sqrt{c + d x^{2}} \left (3 a^{2} d^{2} - 14 a b c d + 35 b^{2} c^{2}\right )}{105 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)**(9/2)/(d*x**2+c)**(1/2),x)

[Out]

a**2*sqrt(c + d*x**2)/(7*b**2*(a + b*x**2)**(7/2)*(a*d - b*c)) - 2*a*sqrt(c + d*
x**2)*(4*a*d - 7*b*c)/(35*b**2*(a + b*x**2)**(5/2)*(a*d - b*c)**2) + 2*d*sqrt(c
+ d*x**2)*(3*a**2*d**2 - 14*a*b*c*d + 35*b**2*c**2)/(105*b**2*sqrt(a + b*x**2)*(
a*d - b*c)**4) + sqrt(c + d*x**2)*(3*a**2*d**2 - 14*a*b*c*d + 35*b**2*c**2)/(105
*b**2*(a + b*x**2)**(3/2)*(a*d - b*c)**3)

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Mathematica [A]  time = 0.215599, size = 151, normalized size = 0.7 \[ \frac{\sqrt{c+d x^2} \left (7 a^3 d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )+a^2 b \left (-8 c^3+200 c^2 d x^2-101 c d^2 x^4+6 d^3 x^6\right )-7 a b^2 c x^2 \left (4 c^2-37 c d x^2+4 d^2 x^4\right )-35 b^3 c^2 x^4 \left (c-2 d x^2\right )\right )}{105 \left (a+b x^2\right )^{7/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^2]*(-35*b^3*c^2*x^4*(c - 2*d*x^2) + 7*a^3*d*(8*c^2 - 4*c*d*x^2 + 3
*d^2*x^4) - 7*a*b^2*c*x^2*(4*c^2 - 37*c*d*x^2 + 4*d^2*x^4) + a^2*b*(-8*c^3 + 200
*c^2*d*x^2 - 101*c*d^2*x^4 + 6*d^3*x^6)))/(105*(b*c - a*d)^4*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.016, size = 213, normalized size = 1. \[{\frac{6\,{a}^{2}b{d}^{3}{x}^{6}-28\,a{b}^{2}c{d}^{2}{x}^{6}+70\,{b}^{3}{c}^{2}d{x}^{6}+21\,{a}^{3}{d}^{3}{x}^{4}-101\,{a}^{2}bc{d}^{2}{x}^{4}+259\,a{b}^{2}{c}^{2}d{x}^{4}-35\,{b}^{3}{c}^{3}{x}^{4}-28\,{a}^{3}c{d}^{2}{x}^{2}+200\,{a}^{2}b{c}^{2}d{x}^{2}-28\,a{b}^{2}{c}^{3}{x}^{2}+56\,{a}^{3}{c}^{2}d-8\,{a}^{2}b{c}^{3}}{105\,{a}^{4}{d}^{4}-420\,{a}^{3}bc{d}^{3}+630\,{a}^{2}{c}^{2}{d}^{2}{b}^{2}-420\,a{c}^{3}d{b}^{3}+105\,{c}^{4}{b}^{4}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^(9/2)/(d*x^2+c)^(1/2),x)

[Out]

1/105*(d*x^2+c)^(1/2)*(6*a^2*b*d^3*x^6-28*a*b^2*c*d^2*x^6+70*b^3*c^2*d*x^6+21*a^
3*d^3*x^4-101*a^2*b*c*d^2*x^4+259*a*b^2*c^2*d*x^4-35*b^3*c^3*x^4-28*a^3*c*d^2*x^
2+200*a^2*b*c^2*d*x^2-28*a*b^2*c^3*x^2+56*a^3*c^2*d-8*a^2*b*c^3)/(b*x^2+a)^(7/2)
/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.494261, size = 609, normalized size = 2.81 \[ \frac{{\left (2 \,{\left (35 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{6} - 8 \, a^{2} b c^{3} + 56 \, a^{3} c^{2} d -{\left (35 \, b^{3} c^{3} - 259 \, a b^{2} c^{2} d + 101 \, a^{2} b c d^{2} - 21 \, a^{3} d^{3}\right )} x^{4} - 4 \,{\left (7 \, a b^{2} c^{3} - 50 \, a^{2} b c^{2} d + 7 \, a^{3} c d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{105 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{8} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{6} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{4} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(2*(35*b^3*c^2*d - 14*a*b^2*c*d^2 + 3*a^2*b*d^3)*x^6 - 8*a^2*b*c^3 + 56*a^
3*c^2*d - (35*b^3*c^3 - 259*a*b^2*c^2*d + 101*a^2*b*c*d^2 - 21*a^3*d^3)*x^4 - 4*
(7*a*b^2*c^3 - 50*a^2*b*c^2*d + 7*a^3*c*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c
)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4 +
(b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x
^8 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*
b^3*d^4)*x^6 + 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*
c*d^3 + a^6*b^2*d^4)*x^4 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2
- 4*a^6*b^2*c*d^3 + a^7*b*d^4)*x^2)

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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(x**5/(b*x**2+a)**(9/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.31608, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done