3.1608 \(\int \frac{1}{(a+b x)^{13/3} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=136 \[ \frac{243 d^3 \sqrt [3]{c+d x}}{140 \sqrt [3]{a+b x} (b c-a d)^4}-\frac{81 d^2 \sqrt [3]{c+d x}}{140 (a+b x)^{4/3} (b c-a d)^3}+\frac{27 d \sqrt [3]{c+d x}}{70 (a+b x)^{7/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{10 (a+b x)^{10/3} (b c-a d)} \]

[Out]

(-3*(c + d*x)^(1/3))/(10*(b*c - a*d)*(a + b*x)^(10/3)) + (27*d*(c + d*x)^(1/3))/
(70*(b*c - a*d)^2*(a + b*x)^(7/3)) - (81*d^2*(c + d*x)^(1/3))/(140*(b*c - a*d)^3
*(a + b*x)^(4/3)) + (243*d^3*(c + d*x)^(1/3))/(140*(b*c - a*d)^4*(a + b*x)^(1/3)
)

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Rubi [A]  time = 0.124892, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{243 d^3 \sqrt [3]{c+d x}}{140 \sqrt [3]{a+b x} (b c-a d)^4}-\frac{81 d^2 \sqrt [3]{c+d x}}{140 (a+b x)^{4/3} (b c-a d)^3}+\frac{27 d \sqrt [3]{c+d x}}{70 (a+b x)^{7/3} (b c-a d)^2}-\frac{3 \sqrt [3]{c+d x}}{10 (a+b x)^{10/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(13/3)*(c + d*x)^(2/3)),x]

[Out]

(-3*(c + d*x)^(1/3))/(10*(b*c - a*d)*(a + b*x)^(10/3)) + (27*d*(c + d*x)^(1/3))/
(70*(b*c - a*d)^2*(a + b*x)^(7/3)) - (81*d^2*(c + d*x)^(1/3))/(140*(b*c - a*d)^3
*(a + b*x)^(4/3)) + (243*d^3*(c + d*x)^(1/3))/(140*(b*c - a*d)^4*(a + b*x)^(1/3)
)

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Rubi in Sympy [A]  time = 19.4857, size = 121, normalized size = 0.89 \[ \frac{243 d^{3} \sqrt [3]{c + d x}}{140 \sqrt [3]{a + b x} \left (a d - b c\right )^{4}} + \frac{81 d^{2} \sqrt [3]{c + d x}}{140 \left (a + b x\right )^{\frac{4}{3}} \left (a d - b c\right )^{3}} + \frac{27 d \sqrt [3]{c + d x}}{70 \left (a + b x\right )^{\frac{7}{3}} \left (a d - b c\right )^{2}} + \frac{3 \sqrt [3]{c + d x}}{10 \left (a + b x\right )^{\frac{10}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(13/3)/(d*x+c)**(2/3),x)

[Out]

243*d**3*(c + d*x)**(1/3)/(140*(a + b*x)**(1/3)*(a*d - b*c)**4) + 81*d**2*(c + d
*x)**(1/3)/(140*(a + b*x)**(4/3)*(a*d - b*c)**3) + 27*d*(c + d*x)**(1/3)/(70*(a
+ b*x)**(7/3)*(a*d - b*c)**2) + 3*(c + d*x)**(1/3)/(10*(a + b*x)**(10/3)*(a*d -
b*c))

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Mathematica [A]  time = 0.240098, size = 95, normalized size = 0.7 \[ \frac{3 \sqrt [3]{c+d x} \left (27 d^2 (a+b x)^2 (a d-b c)+18 d (a+b x) (b c-a d)^2-14 (b c-a d)^3+81 d^3 (a+b x)^3\right )}{140 (a+b x)^{10/3} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(13/3)*(c + d*x)^(2/3)),x]

[Out]

(3*(c + d*x)^(1/3)*(-14*(b*c - a*d)^3 + 18*d*(b*c - a*d)^2*(a + b*x) + 27*d^2*(-
(b*c) + a*d)*(a + b*x)^2 + 81*d^3*(a + b*x)^3))/(140*(b*c - a*d)^4*(a + b*x)^(10
/3))

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Maple [A]  time = 0.014, size = 171, normalized size = 1.3 \[{\frac{243\,{b}^{3}{d}^{3}{x}^{3}+810\,a{b}^{2}{d}^{3}{x}^{2}-81\,{b}^{3}c{d}^{2}{x}^{2}+945\,{a}^{2}b{d}^{3}x-270\,a{b}^{2}c{d}^{2}x+54\,{b}^{3}{c}^{2}dx+420\,{a}^{3}{d}^{3}-315\,{a}^{2}cb{d}^{2}+180\,a{b}^{2}{c}^{2}d-42\,{b}^{3}{c}^{3}}{140\,{d}^{4}{a}^{4}-560\,b{d}^{3}c{a}^{3}+840\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-560\,{b}^{3}d{c}^{3}a+140\,{b}^{4}{c}^{4}}\sqrt [3]{dx+c} \left ( bx+a \right ) ^{-{\frac{10}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(13/3)/(d*x+c)^(2/3),x)

[Out]

3/140*(d*x+c)^(1/3)*(81*b^3*d^3*x^3+270*a*b^2*d^3*x^2-27*b^3*c*d^2*x^2+315*a^2*b
*d^3*x-90*a*b^2*c*d^2*x+18*b^3*c^2*d*x+140*a^3*d^3-105*a^2*b*c*d^2+60*a*b^2*c^2*
d-14*b^3*c^3)/(b*x+a)^(10/3)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^
3*d+b^4*c^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{13}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(13/3)*(d*x + c)^(2/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(13/3)*(d*x + c)^(2/3)), x)

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Fricas [A]  time = 0.214053, size = 566, normalized size = 4.16 \[ \frac{3 \,{\left (81 \, b^{3} d^{3} x^{3} - 14 \, b^{3} c^{3} + 60 \, a b^{2} c^{2} d - 105 \, a^{2} b c d^{2} + 140 \, a^{3} d^{3} - 27 \,{\left (b^{3} c d^{2} - 10 \, a b^{2} d^{3}\right )} x^{2} + 9 \,{\left (2 \, b^{3} c^{2} d - 10 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{140 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(13/3)*(d*x + c)^(2/3)),x, algorithm="fricas")

[Out]

3/140*(81*b^3*d^3*x^3 - 14*b^3*c^3 + 60*a*b^2*c^2*d - 105*a^2*b*c*d^2 + 140*a^3*
d^3 - 27*(b^3*c*d^2 - 10*a*b^2*d^3)*x^2 + 9*(2*b^3*c^2*d - 10*a*b^2*c*d^2 + 35*a
^2*b*d^3)*x)*(b*x + a)^(2/3)*(d*x + c)^(1/3)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*
a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6
*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^4 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d +
6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^3 + 6*(a^2*b^6*c^4 - 4*a^3
*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^2 + 4*(a^3*b^5
*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4)*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)**(13/3)/(d*x+c)**(2/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{13}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(13/3)*(d*x + c)^(2/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(13/3)*(d*x + c)^(2/3)), x)