3.1829 \(\int \frac{1}{(a+b x)^{5/6} (c+d x)^{13/6}} \, dx\)

Optimal. Leaf size=66 \[ \frac{36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac{6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(1/6))/(7*(b*c - a*d)*(c + d*x)^(7/6)) + (36*b*(a + b*x)^(1/6))/(7*
(b*c - a*d)^2*(c + d*x)^(1/6))

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Rubi [A]  time = 0.0517297, antiderivative size = 66, 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 \[ \frac{36 b \sqrt [6]{a+b x}}{7 \sqrt [6]{c+d x} (b c-a d)^2}+\frac{6 \sqrt [6]{a+b x}}{7 (c+d x)^{7/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/6)*(c + d*x)^(13/6)),x]

[Out]

(6*(a + b*x)^(1/6))/(7*(b*c - a*d)*(c + d*x)^(7/6)) + (36*b*(a + b*x)^(1/6))/(7*
(b*c - a*d)^2*(c + d*x)^(1/6))

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Rubi in Sympy [A]  time = 6.91437, size = 56, normalized size = 0.85 \[ \frac{36 b \sqrt [6]{a + b x}}{7 \sqrt [6]{c + d x} \left (a d - b c\right )^{2}} - \frac{6 \sqrt [6]{a + b x}}{7 \left (c + d x\right )^{\frac{7}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/6)/(d*x+c)**(13/6),x)

[Out]

36*b*(a + b*x)**(1/6)/(7*(c + d*x)**(1/6)*(a*d - b*c)**2) - 6*(a + b*x)**(1/6)/(
7*(c + d*x)**(7/6)*(a*d - b*c))

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Mathematica [A]  time = 0.0600439, size = 46, normalized size = 0.7 \[ \frac{6 \sqrt [6]{a+b x} (-a d+7 b c+6 b d x)}{7 (c+d x)^{7/6} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/6)*(c + d*x)^(13/6)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 53, normalized size = 0.8 \[ -{\frac{-36\,bdx+6\,ad-42\,bc}{7\,{a}^{2}{d}^{2}-14\,abcd+7\,{b}^{2}{c}^{2}}\sqrt [6]{bx+a} \left ( dx+c \right ) ^{-{\frac{7}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/6)/(d*x+c)^(13/6),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{13}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/6)*(d*x + c)^(13/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(13/6)), x)

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Fricas [A]  time = 0.213918, size = 159, normalized size = 2.41 \[ \frac{6 \,{\left (6 \, b d x + 7 \, b c - a d\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{7 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/6)*(d*x + c)^(13/6)),x, algorithm="fricas")

[Out]

6/7*(6*b*d*x + 7*b*c - a*d)*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b^2*c^4 - 2*a*b*c^3
*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*
a*b*c^2*d^2 + a^2*c*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)**(5/6)/(d*x+c)**(13/6),x)

[Out]

Timed out

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GIAC/XCAS [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)^(5/6)*(d*x + c)^(13/6)),x, algorithm="giac")

[Out]

Timed out