3.1828 \(\int \frac{1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx\)

Optimal. Leaf size=30 \[ \frac{6 \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.022563, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{6 \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/6)*(c + d*x)^(7/6)),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.32563, size = 26, normalized size = 0.87 \[ - \frac{6 \sqrt [6]{a + b x}}{\sqrt [6]{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/6)/(d*x+c)**(7/6),x)

[Out]

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

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Mathematica [A]  time = 0.0385026, size = 30, normalized size = 1. \[ -\frac{6 \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/6)*(c + d*x)^(7/6)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 27, normalized size = 0.9 \[ -6\,{\frac{\sqrt [6]{bx+a}}{\sqrt [6]{dx+c} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/6)/(d*x+c)^(7/6),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{6}}{\left (d x + c\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/6)*(d*x + c)^(7/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/6)*(d*x + c)^(7/6)), x)

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Fricas [A]  time = 0.214111, size = 57, normalized size = 1.9 \[ \frac{6 \,{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/6)*(d*x + c)^(7/6)),x, algorithm="fricas")

[Out]

6*(b*x + a)^(1/6)*(d*x + c)^(5/6)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*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)**(7/6),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)^(7/6)),x, algorithm="giac")

[Out]

Timed out