3.1607 \(\int \frac{1}{(a+b x)^{10/3} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{27 d^2 \sqrt [3]{c+d x}}{14 \sqrt [3]{a+b x} (b c-a d)^3}+\frac{9 d \sqrt [3]{c+d x}}{14 (a+b x)^{4/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{7 (a+b x)^{7/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(1/3))/(7*(b*c - a*d)*(a + b*x)^(7/3)) + (9*d*(c + d*x)^(1/3))/(14
*(b*c - a*d)^2*(a + b*x)^(4/3)) - (27*d^2*(c + d*x)^(1/3))/(14*(b*c - a*d)^3*(a
+ b*x)^(1/3))

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Rubi [A]  time = 0.0856323, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{27 d^2 \sqrt [3]{c+d x}}{14 \sqrt [3]{a+b x} (b c-a d)^3}+\frac{9 d \sqrt [3]{c+d x}}{14 (a+b x)^{4/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{7 (a+b x)^{7/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(10/3)*(c + d*x)^(2/3)),x]

[Out]

(-3*(c + d*x)^(1/3))/(7*(b*c - a*d)*(a + b*x)^(7/3)) + (9*d*(c + d*x)^(1/3))/(14
*(b*c - a*d)^2*(a + b*x)^(4/3)) - (27*d^2*(c + d*x)^(1/3))/(14*(b*c - a*d)^3*(a
+ b*x)^(1/3))

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Rubi in Sympy [A]  time = 12.8044, size = 88, normalized size = 0.87 \[ \frac{27 d^{2} \sqrt [3]{c + d x}}{14 \sqrt [3]{a + b x} \left (a d - b c\right )^{3}} + \frac{9 d \sqrt [3]{c + d x}}{14 \left (a + b x\right )^{\frac{4}{3}} \left (a d - b c\right )^{2}} + \frac{3 \sqrt [3]{c + d x}}{7 \left (a + b x\right )^{\frac{7}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(10/3)/(d*x+c)**(2/3),x)

[Out]

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

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Mathematica [A]  time = 0.092654, size = 75, normalized size = 0.74 \[ -\frac{3 \sqrt [3]{c+d x} \left (14 a^2 d^2-7 a b d (c-3 d x)+b^2 \left (2 c^2-3 c d x+9 d^2 x^2\right )\right )}{14 (a+b x)^{7/3} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(10/3)*(c + d*x)^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[{\frac{27\,{b}^{2}{d}^{2}{x}^{2}+63\,ab{d}^{2}x-9\,{b}^{2}cdx+42\,{a}^{2}{d}^{2}-21\,abcd+6\,{b}^{2}{c}^{2}}{14\,{a}^{3}{d}^{3}-42\,{a}^{2}cb{d}^{2}+42\,a{b}^{2}{c}^{2}d-14\,{b}^{3}{c}^{3}}\sqrt [3]{dx+c} \left ( bx+a \right ) ^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(10/3)/(d*x+c)^(2/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{10}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(10/3)*(d*x + c)^(2/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(10/3)*(d*x + c)^(2/3)), x)

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Fricas [A]  time = 0.209761, size = 339, normalized size = 3.36 \[ -\frac{3 \,{\left (9 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 14 \, a^{2} d^{2} - 3 \,{\left (b^{2} c d - 7 \, a b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{14 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(10/3)*(d*x + c)^(2/3)),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{10}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(10/3)*(d*x + c)^(2/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(10/3)*(d*x + c)^(2/3)), x)