3.1810 \(\int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx\)

Optimal. Leaf size=66 \[ \frac{36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(5/6))/(11*(b*c - a*d)*(c + d*x)^(11/6)) + (36*b*(a + b*x)^(5/6))/(
55*(b*c - a*d)^2*(c + d*x)^(5/6))

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Rubi [A]  time = 0.0511038, 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 \[ \frac{36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(1/6)*(c + d*x)^(17/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(11*(b*c - a*d)*(c + d*x)^(11/6)) + (36*b*(a + b*x)^(5/6))/(
55*(b*c - a*d)^2*(c + d*x)^(5/6))

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Rubi in Sympy [A]  time = 6.9303, size = 56, normalized size = 0.85 \[ \frac{36 b \left (a + b x\right )^{\frac{5}{6}}}{55 \left (c + d x\right )^{\frac{5}{6}} \left (a d - b c\right )^{2}} - \frac{6 \left (a + b x\right )^{\frac{5}{6}}}{11 \left (c + d x\right )^{\frac{11}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/6)/(d*x+c)**(17/6),x)

[Out]

36*b*(a + b*x)**(5/6)/(55*(c + d*x)**(5/6)*(a*d - b*c)**2) - 6*(a + b*x)**(5/6)/
(11*(c + d*x)**(11/6)*(a*d - b*c))

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Mathematica [A]  time = 0.0641339, size = 46, normalized size = 0.7 \[ \frac{6 (a+b x)^{5/6} (-5 a d+11 b c+6 b d x)}{55 (c+d x)^{11/6} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(17/6)),x]

[Out]

(6*(a + b*x)^(5/6)*(11*b*c - 5*a*d + 6*b*d*x))/(55*(b*c - a*d)^2*(c + d*x)^(11/6
))

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \[ -{\frac{-36\,bdx+30\,ad-66\,bc}{55\,{a}^{2}{d}^{2}-110\,abcd+55\,{b}^{2}{c}^{2}} \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{11}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/6)/(d*x+c)^(17/6),x)

[Out]

-6/55*(b*x+a)^(5/6)*(-6*b*d*x+5*a*d-11*b*c)/(d*x+c)^(11/6)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{17}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(17/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(17/6)), x)

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Fricas [A]  time = 0.228813, size = 138, normalized size = 2.09 \[ \frac{6 \,{\left (6 \, b^{2} d x^{2} + 11 \, a b c - 5 \, a^{2} d +{\left (11 \, b^{2} c + a b d\right )} x\right )}}{55 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(17/6)),x, algorithm="fricas")

[Out]

6/55*(6*b^2*d*x^2 + 11*a*b*c - 5*a^2*d + (11*b^2*c + a*b*d)*x)/((b^2*c^3 - 2*a*b
*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x)*(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)**(1/6)/(d*x+c)**(17/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{17}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(17/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(17/6)), x)