3.369 \(\int x^7 \left (a+b x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^4}-\frac{a \left (a+b x^2\right )^{9/2}}{3 b^4} \]

[Out]

-(a^3*(a + b*x^2)^(5/2))/(5*b^4) + (3*a^2*(a + b*x^2)^(7/2))/(7*b^4) - (a*(a + b
*x^2)^(9/2))/(3*b^4) + (a + b*x^2)^(11/2)/(11*b^4)

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Rubi [A]  time = 0.11816, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac{\left (a+b x^2\right )^{11/2}}{11 b^4}-\frac{a \left (a+b x^2\right )^{9/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x^2)^(3/2),x]

[Out]

-(a^3*(a + b*x^2)^(5/2))/(5*b^4) + (3*a^2*(a + b*x^2)^(7/2))/(7*b^4) - (a*(a + b
*x^2)^(9/2))/(3*b^4) + (a + b*x^2)^(11/2)/(11*b^4)

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Rubi in Sympy [A]  time = 15.6366, size = 70, normalized size = 0.88 \[ - \frac{a^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{3 a^{2} \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{4}} - \frac{a \left (a + b x^{2}\right )^{\frac{9}{2}}}{3 b^{4}} + \frac{\left (a + b x^{2}\right )^{\frac{11}{2}}}{11 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x**2+a)**(3/2),x)

[Out]

-a**3*(a + b*x**2)**(5/2)/(5*b**4) + 3*a**2*(a + b*x**2)**(7/2)/(7*b**4) - a*(a
+ b*x**2)**(9/2)/(3*b**4) + (a + b*x**2)**(11/2)/(11*b**4)

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Mathematica [A]  time = 0.0446056, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^2\right )^{5/2} \left (-16 a^3+40 a^2 b x^2-70 a b^2 x^4+105 b^3 x^6\right )}{1155 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x^2)^(3/2),x]

[Out]

((a + b*x^2)^(5/2)*(-16*a^3 + 40*a^2*b*x^2 - 70*a*b^2*x^4 + 105*b^3*x^6))/(1155*
b^4)

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Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-105\,{b}^{3}{x}^{6}+70\,a{b}^{2}{x}^{4}-40\,{a}^{2}b{x}^{2}+16\,{a}^{3}}{1155\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^2+a)^(3/2),x)

[Out]

-1/1155*(b*x^2+a)^(5/2)*(-105*b^3*x^6+70*a*b^2*x^4-40*a^2*b*x^2+16*a^3)/b^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((b*x^2 + a)^(3/2)*x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237386, size = 92, normalized size = 1.15 \[ \frac{{\left (105 \, b^{5} x^{10} + 140 \, a b^{4} x^{8} + 5 \, a^{2} b^{3} x^{6} - 6 \, a^{3} b^{2} x^{4} + 8 \, a^{4} b x^{2} - 16 \, a^{5}\right )} \sqrt{b x^{2} + a}}{1155 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^7,x, algorithm="fricas")

[Out]

1/1155*(105*b^5*x^10 + 140*a*b^4*x^8 + 5*a^2*b^3*x^6 - 6*a^3*b^2*x^4 + 8*a^4*b*x
^2 - 16*a^5)*sqrt(b*x^2 + a)/b^4

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Sympy [A]  time = 11.7839, size = 133, normalized size = 1.66 \[ \begin{cases} - \frac{16 a^{5} \sqrt{a + b x^{2}}}{1155 b^{4}} + \frac{8 a^{4} x^{2} \sqrt{a + b x^{2}}}{1155 b^{3}} - \frac{2 a^{3} x^{4} \sqrt{a + b x^{2}}}{385 b^{2}} + \frac{a^{2} x^{6} \sqrt{a + b x^{2}}}{231 b} + \frac{4 a x^{8} \sqrt{a + b x^{2}}}{33} + \frac{b x^{10} \sqrt{a + b x^{2}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{8}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x**2+a)**(3/2),x)

[Out]

Piecewise((-16*a**5*sqrt(a + b*x**2)/(1155*b**4) + 8*a**4*x**2*sqrt(a + b*x**2)/
(1155*b**3) - 2*a**3*x**4*sqrt(a + b*x**2)/(385*b**2) + a**2*x**6*sqrt(a + b*x**
2)/(231*b) + 4*a*x**8*sqrt(a + b*x**2)/33 + b*x**10*sqrt(a + b*x**2)/11, Ne(b, 0
)), (a**(3/2)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.209992, size = 181, normalized size = 2.26 \[ \frac{\frac{11 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} a}{b^{3}} + \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}}{b^{3}}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^7,x, algorithm="giac")

[Out]

1/3465*(11*(35*(b*x^2 + a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/
2)*a^2 - 105*(b*x^2 + a)^(3/2)*a^3)*a/b^3 + (315*(b*x^2 + a)^(11/2) - 1540*(b*x^
2 + a)^(9/2)*a + 2970*(b*x^2 + a)^(7/2)*a^2 - 2772*(b*x^2 + a)^(5/2)*a^3 + 1155*
(b*x^2 + a)^(3/2)*a^4)/b^3)/b