3.690 \(\int x^5 \left (a+b x^2\right )^{4/3} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^3}-\frac{3 a \left (a+b x^2\right )^{10/3}}{10 b^3} \]

[Out]

(3*a^2*(a + b*x^2)^(7/3))/(14*b^3) - (3*a*(a + b*x^2)^(10/3))/(10*b^3) + (3*(a +
b*x^2)^(13/3))/(26*b^3)

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Rubi [A]  time = 0.102202, antiderivative size = 59, 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 \[ \frac{3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}+\frac{3 \left (a+b x^2\right )^{13/3}}{26 b^3}-\frac{3 a \left (a+b x^2\right )^{10/3}}{10 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^(4/3),x]

[Out]

(3*a^2*(a + b*x^2)^(7/3))/(14*b^3) - (3*a*(a + b*x^2)^(10/3))/(10*b^3) + (3*(a +
b*x^2)^(13/3))/(26*b^3)

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Rubi in Sympy [A]  time = 11.6929, size = 54, normalized size = 0.92 \[ \frac{3 a^{2} \left (a + b x^{2}\right )^{\frac{7}{3}}}{14 b^{3}} - \frac{3 a \left (a + b x^{2}\right )^{\frac{10}{3}}}{10 b^{3}} + \frac{3 \left (a + b x^{2}\right )^{\frac{13}{3}}}{26 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**(4/3),x)

[Out]

3*a**2*(a + b*x**2)**(7/3)/(14*b**3) - 3*a*(a + b*x**2)**(10/3)/(10*b**3) + 3*(a
+ b*x**2)**(13/3)/(26*b**3)

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Mathematica [A]  time = 0.0342535, size = 39, normalized size = 0.66 \[ \frac{3 \left (a+b x^2\right )^{7/3} \left (9 a^2-21 a b x^2+35 b^2 x^4\right )}{910 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^(4/3),x]

[Out]

(3*(a + b*x^2)^(7/3)*(9*a^2 - 21*a*b*x^2 + 35*b^2*x^4))/(910*b^3)

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Maple [A]  time = 0.007, size = 36, normalized size = 0.6 \[{\frac{105\,{b}^{2}{x}^{4}-63\,ab{x}^{2}+27\,{a}^{2}}{910\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^(4/3),x)

[Out]

3/910*(b*x^2+a)^(7/3)*(35*b^2*x^4-21*a*b*x^2+9*a^2)/b^3

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Maxima [A]  time = 1.34555, size = 63, normalized size = 1.07 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}}}{26 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a}{10 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2}}{14 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)*x^5,x, algorithm="maxima")

[Out]

3/26*(b*x^2 + a)^(13/3)/b^3 - 3/10*(b*x^2 + a)^(10/3)*a/b^3 + 3/14*(b*x^2 + a)^(
7/3)*a^2/b^3

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Fricas [A]  time = 0.208169, size = 77, normalized size = 1.31 \[ \frac{3 \,{\left (35 \, b^{4} x^{8} + 49 \, a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{4} - 3 \, a^{3} b x^{2} + 9 \, a^{4}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{910 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)*x^5,x, algorithm="fricas")

[Out]

3/910*(35*b^4*x^8 + 49*a*b^3*x^6 + 2*a^2*b^2*x^4 - 3*a^3*b*x^2 + 9*a^4)*(b*x^2 +
a)^(1/3)/b^3

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Sympy [A]  time = 12.5168, size = 112, normalized size = 1.9 \[ \begin{cases} \frac{27 a^{4} \sqrt [3]{a + b x^{2}}}{910 b^{3}} - \frac{9 a^{3} x^{2} \sqrt [3]{a + b x^{2}}}{910 b^{2}} + \frac{3 a^{2} x^{4} \sqrt [3]{a + b x^{2}}}{455 b} + \frac{21 a x^{6} \sqrt [3]{a + b x^{2}}}{130} + \frac{3 b x^{8} \sqrt [3]{a + b x^{2}}}{26} & \text{for}\: b \neq 0 \\\frac{a^{\frac{4}{3}} x^{6}}{6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**(4/3),x)

[Out]

Piecewise((27*a**4*(a + b*x**2)**(1/3)/(910*b**3) - 9*a**3*x**2*(a + b*x**2)**(1
/3)/(910*b**2) + 3*a**2*x**4*(a + b*x**2)**(1/3)/(455*b) + 21*a*x**6*(a + b*x**2
)**(1/3)/130 + 3*b*x**8*(a + b*x**2)**(1/3)/26, Ne(b, 0)), (a**(4/3)*x**6/6, Tru
e))

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GIAC/XCAS [A]  time = 0.214999, size = 143, normalized size = 2.42 \[ \frac{3 \,{\left (\frac{13 \,{\left (14 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{2}\right )} a}{b^{2}} + \frac{140 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{3}}{b^{2}}\right )}}{3640 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)*x^5,x, algorithm="giac")

[Out]

3/3640*(13*(14*(b*x^2 + a)^(10/3) - 40*(b*x^2 + a)^(7/3)*a + 35*(b*x^2 + a)^(4/3
)*a^2)*a/b^2 + (140*(b*x^2 + a)^(13/3) - 546*(b*x^2 + a)^(10/3)*a + 780*(b*x^2 +
a)^(7/3)*a^2 - 455*(b*x^2 + a)^(4/3)*a^3)/b^2)/b