3.28 \(\int \frac{\left (a+b x^3\right )^2}{\left (c+d x^3\right )^{10/3}} \, dx\)

Optimal. Leaf size=78 \[ \frac{9 a^2 x}{14 c^3 \sqrt [3]{c+d x^3}}+\frac{3 a x \left (a+b x^3\right )}{14 c^2 \left (c+d x^3\right )^{4/3}}+\frac{x \left (a+b x^3\right )^2}{7 c \left (c+d x^3\right )^{7/3}} \]

[Out]

(x*(a + b*x^3)^2)/(7*c*(c + d*x^3)^(7/3)) + (3*a*x*(a + b*x^3))/(14*c^2*(c + d*x
^3)^(4/3)) + (9*a^2*x)/(14*c^3*(c + d*x^3)^(1/3))

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Rubi [A]  time = 0.0701422, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{9 a^2 x}{14 c^3 \sqrt [3]{c+d x^3}}+\frac{3 a x \left (a+b x^3\right )}{14 c^2 \left (c+d x^3\right )^{4/3}}+\frac{x \left (a+b x^3\right )^2}{7 c \left (c+d x^3\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2/(c + d*x^3)^(10/3),x]

[Out]

(x*(a + b*x^3)^2)/(7*c*(c + d*x^3)^(7/3)) + (3*a*x*(a + b*x^3))/(14*c^2*(c + d*x
^3)^(4/3)) + (9*a^2*x)/(14*c^3*(c + d*x^3)^(1/3))

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Rubi in Sympy [A]  time = 11.2865, size = 71, normalized size = 0.91 \[ \frac{9 a^{2} x}{14 c^{3} \sqrt [3]{c + d x^{3}}} + \frac{3 a x \left (a + b x^{3}\right )}{14 c^{2} \left (c + d x^{3}\right )^{\frac{4}{3}}} + \frac{x \left (a + b x^{3}\right )^{2}}{7 c \left (c + d x^{3}\right )^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2/(d*x**3+c)**(10/3),x)

[Out]

9*a**2*x/(14*c**3*(c + d*x**3)**(1/3)) + 3*a*x*(a + b*x**3)/(14*c**2*(c + d*x**3
)**(4/3)) + x*(a + b*x**3)**2/(7*c*(c + d*x**3)**(7/3))

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Mathematica [A]  time = 0.0835213, size = 73, normalized size = 0.94 \[ \frac{a^2 \left (14 c^2 x+21 c d x^4+9 d^2 x^7\right )+a b c x^4 \left (7 c+3 d x^3\right )+2 b^2 c^2 x^7}{14 c^3 \left (c+d x^3\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2/(c + d*x^3)^(10/3),x]

[Out]

(2*b^2*c^2*x^7 + a*b*c*x^4*(7*c + 3*d*x^3) + a^2*(14*c^2*x + 21*c*d*x^4 + 9*d^2*
x^7))/(14*c^3*(c + d*x^3)^(7/3))

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Maple [A]  time = 0.01, size = 76, normalized size = 1. \[{\frac{x \left ( 9\,{a}^{2}{d}^{2}{x}^{6}+3\,abcd{x}^{6}+2\,{b}^{2}{c}^{2}{x}^{6}+21\,{a}^{2}cd{x}^{3}+7\,ab{c}^{2}{x}^{3}+14\,{a}^{2}{c}^{2} \right ) }{14\,{c}^{3}} \left ( d{x}^{3}+c \right ) ^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/(d*x^3+c)^(10/3),x)

[Out]

1/14*x*(9*a^2*d^2*x^6+3*a*b*c*d*x^6+2*b^2*c^2*x^6+21*a^2*c*d*x^3+7*a*b*c^2*x^3+1
4*a^2*c^2)/(d*x^3+c)^(7/3)/c^3

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Maxima [A]  time = 1.38355, size = 147, normalized size = 1.88 \[ \frac{b^{2} x^{7}}{7 \,{\left (d x^{3} + c\right )}^{\frac{7}{3}} c} - \frac{a b{\left (4 \, d - \frac{7 \,{\left (d x^{3} + c\right )}}{x^{3}}\right )} x^{7}}{14 \,{\left (d x^{3} + c\right )}^{\frac{7}{3}} c^{2}} + \frac{{\left (2 \, d^{2} - \frac{7 \,{\left (d x^{3} + c\right )} d}{x^{3}} + \frac{14 \,{\left (d x^{3} + c\right )}^{2}}{x^{6}}\right )} a^{2} x^{7}}{14 \,{\left (d x^{3} + c\right )}^{\frac{7}{3}} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/(d*x^3 + c)^(10/3),x, algorithm="maxima")

[Out]

1/7*b^2*x^7/((d*x^3 + c)^(7/3)*c) - 1/14*a*b*(4*d - 7*(d*x^3 + c)/x^3)*x^7/((d*x
^3 + c)^(7/3)*c^2) + 1/14*(2*d^2 - 7*(d*x^3 + c)*d/x^3 + 14*(d*x^3 + c)^2/x^6)*a
^2*x^7/((d*x^3 + c)^(7/3)*c^3)

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Fricas [A]  time = 0.21906, size = 139, normalized size = 1.78 \[ \frac{{\left ({\left (2 \, b^{2} c^{2} + 3 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{7} + 14 \, a^{2} c^{2} x + 7 \,{\left (a b c^{2} + 3 \, a^{2} c d\right )} x^{4}\right )}{\left (d x^{3} + c\right )}^{\frac{2}{3}}}{14 \,{\left (c^{3} d^{3} x^{9} + 3 \, c^{4} d^{2} x^{6} + 3 \, c^{5} d x^{3} + c^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/(d*x^3 + c)^(10/3),x, algorithm="fricas")

[Out]

1/14*((2*b^2*c^2 + 3*a*b*c*d + 9*a^2*d^2)*x^7 + 14*a^2*c^2*x + 7*(a*b*c^2 + 3*a^
2*c*d)*x^4)*(d*x^3 + c)^(2/3)/(c^3*d^3*x^9 + 3*c^4*d^2*x^6 + 3*c^5*d*x^3 + c^6)

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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((b*x**3+a)**2/(d*x**3+c)**(10/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{2}}{{\left (d x^{3} + c\right )}^{\frac{10}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/(d*x^3 + c)^(10/3),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^2/(d*x^3 + c)^(10/3), x)