3.1836 \(\int \frac{1}{(a+b x)^{7/6} (c+d x)^{11/6}} \, dx\)

Optimal. Leaf size=64 \[ -\frac{36 d (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{5/6} (b c-a d)} \]

[Out]

-6/((b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(5/6)) - (36*d*(a + b*x)^(5/6))/(5*(b*
c - a*d)^2*(c + d*x)^(5/6))

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Rubi [A]  time = 0.0535162, antiderivative size = 64, 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 \[ -\frac{36 d (a+b x)^{5/6}}{5 (c+d x)^{5/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{5/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(7/6)*(c + d*x)^(11/6)),x]

[Out]

-6/((b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(5/6)) - (36*d*(a + b*x)^(5/6))/(5*(b*
c - a*d)^2*(c + d*x)^(5/6))

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Rubi in Sympy [A]  time = 6.79483, size = 54, normalized size = 0.84 \[ - \frac{36 d \left (a + b x\right )^{\frac{5}{6}}}{5 \left (c + d x\right )^{\frac{5}{6}} \left (a d - b c\right )^{2}} + \frac{6}{\sqrt [6]{a + b x} \left (c + d x\right )^{\frac{5}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(7/6)/(d*x+c)**(11/6),x)

[Out]

-36*d*(a + b*x)**(5/6)/(5*(c + d*x)**(5/6)*(a*d - b*c)**2) + 6/((a + b*x)**(1/6)
*(c + d*x)**(5/6)*(a*d - b*c))

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Mathematica [A]  time = 0.0686956, size = 45, normalized size = 0.7 \[ -\frac{6 (a d+5 b c+6 b d x)}{5 \sqrt [6]{a+b x} (c+d x)^{5/6} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(11/6)),x]

[Out]

(-6*(5*b*c + a*d + 6*b*d*x))/(5*(b*c - a*d)^2*(a + b*x)^(1/6)*(c + d*x)^(5/6))

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Maple [A]  time = 0.009, size = 53, normalized size = 0.8 \[ -{\frac{36\,bdx+6\,ad+30\,bc}{5\,{a}^{2}{d}^{2}-10\,abcd+5\,{b}^{2}{c}^{2}}{\frac{1}{\sqrt [6]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{5}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(7/6)/(d*x+c)^(11/6),x)

[Out]

-6/5*(6*b*d*x+a*d+5*b*c)/(b*x+a)^(1/6)/(d*x+c)^(5/6)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{11}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(11/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(11/6)), x)

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Fricas [A]  time = 0.21362, size = 70, normalized size = 1.09 \[ -\frac{6 \,{\left (6 \, b d x + 5 \, b c + a d\right )}}{5 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(11/6)),x, algorithm="fricas")

[Out]

-6/5*(6*b*d*x + 5*b*c + a*d)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(1/6)*(d
*x + c)^(5/6))

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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(1/(b*x+a)**(7/6)/(d*x+c)**(11/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{11}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(11/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(11/6)), x)