3.29 \(\int \frac{a+b x^3}{\left (c+d x^3\right )^{7/3}} \, dx\)

Optimal. Leaf size=47 \[ \frac{x \left (a+b x^3\right )}{4 c \left (c+d x^3\right )^{4/3}}+\frac{3 a x}{4 c^2 \sqrt [3]{c+d x^3}} \]

[Out]

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

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Rubi [A]  time = 0.0364384, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (a+b x^3\right )}{4 c \left (c+d x^3\right )^{4/3}}+\frac{3 a x}{4 c^2 \sqrt [3]{c+d x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)/(c + d*x^3)^(7/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.49093, size = 41, normalized size = 0.87 \[ \frac{3 a x}{4 c^{2} \sqrt [3]{c + d x^{3}}} + \frac{x \left (a + b x^{3}\right )}{4 c \left (c + d x^{3}\right )^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.044828, size = 37, normalized size = 0.79 \[ \frac{x \left (4 a c+3 a d x^3+b c x^3\right )}{4 c^2 \left (c+d x^3\right )^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(4*a*c + b*c*x^3 + 3*a*d*x^3))/(4*c^2*(c + d*x^3)^(4/3))

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Maple [A]  time = 0.005, size = 34, normalized size = 0.7 \[{\frac{x \left ( 3\,ad{x}^{3}+bc{x}^{3}+4\,ac \right ) }{4\,{c}^{2}} \left ( d{x}^{3}+c \right ) ^{-{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)/(d*x^3+c)^(7/3),x)

[Out]

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

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Maxima [A]  time = 1.36086, size = 69, normalized size = 1.47 \[ \frac{b x^{4}}{4 \,{\left (d x^{3} + c\right )}^{\frac{4}{3}} c} - \frac{a{\left (d - \frac{4 \,{\left (d x^{3} + c\right )}}{x^{3}}\right )} x^{4}}{4 \,{\left (d x^{3} + c\right )}^{\frac{4}{3}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)/(d*x^3 + c)^(7/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.217738, size = 73, normalized size = 1.55 \[ \frac{{\left ({\left (b c + 3 \, a d\right )} x^{4} + 4 \, a c x\right )}{\left (d x^{3} + c\right )}^{\frac{2}{3}}}{4 \,{\left (c^{2} d^{2} x^{6} + 2 \, c^{3} d x^{3} + c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)/(d*x**3+c)**(7/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{3} + a}{{\left (d x^{3} + c\right )}^{\frac{7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)/(d*x^3 + c)^(7/3), x)