3.1811 \(\int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{23/6}} \, dx\)

Optimal. Leaf size=101 \[ \frac{432 b^2 (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^3}+\frac{72 b (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(5/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (72*b*(a + b*x)^(5/6))/(
187*(b*c - a*d)^2*(c + d*x)^(11/6)) + (432*b^2*(a + b*x)^(5/6))/(935*(b*c - a*d)
^3*(c + d*x)^(5/6))

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Rubi [A]  time = 0.0843786, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{432 b^2 (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^3}+\frac{72 b (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(17*(b*c - a*d)*(c + d*x)^(17/6)) + (72*b*(a + b*x)^(5/6))/(
187*(b*c - a*d)^2*(c + d*x)^(11/6)) + (432*b^2*(a + b*x)^(5/6))/(935*(b*c - a*d)
^3*(c + d*x)^(5/6))

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Rubi in Sympy [A]  time = 12.4571, size = 88, normalized size = 0.87 \[ - \frac{432 b^{2} \left (a + b x\right )^{\frac{5}{6}}}{935 \left (c + d x\right )^{\frac{5}{6}} \left (a d - b c\right )^{3}} + \frac{72 b \left (a + b x\right )^{\frac{5}{6}}}{187 \left (c + d x\right )^{\frac{11}{6}} \left (a d - b c\right )^{2}} - \frac{6 \left (a + b x\right )^{\frac{5}{6}}}{17 \left (c + d x\right )^{\frac{17}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/6)/(d*x+c)**(23/6),x)

[Out]

-432*b**2*(a + b*x)**(5/6)/(935*(c + d*x)**(5/6)*(a*d - b*c)**3) + 72*b*(a + b*x
)**(5/6)/(187*(c + d*x)**(11/6)*(a*d - b*c)**2) - 6*(a + b*x)**(5/6)/(17*(c + d*
x)**(17/6)*(a*d - b*c))

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Mathematica [A]  time = 0.098668, size = 77, normalized size = 0.76 \[ \frac{6 (a+b x)^{5/6} \left (55 a^2 d^2-10 a b d (17 c+6 d x)+b^2 \left (187 c^2+204 c d x+72 d^2 x^2\right )\right )}{935 (c+d x)^{17/6} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(5/6)*(55*a^2*d^2 - 10*a*b*d*(17*c + 6*d*x) + b^2*(187*c^2 + 204*c*
d*x + 72*d^2*x^2)))/(935*(b*c - a*d)^3*(c + d*x)^(17/6))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[ -{\frac{432\,{b}^{2}{d}^{2}{x}^{2}-360\,ab{d}^{2}x+1224\,{b}^{2}cdx+330\,{a}^{2}{d}^{2}-1020\,abcd+1122\,{b}^{2}{c}^{2}}{935\,{a}^{3}{d}^{3}-2805\,{a}^{2}cb{d}^{2}+2805\,a{b}^{2}{c}^{2}d-935\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{17}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/6)/(d*x+c)^(23/6),x)

[Out]

-6/935*(b*x+a)^(5/6)*(72*b^2*d^2*x^2-60*a*b*d^2*x+204*b^2*c*d*x+55*a^2*d^2-170*a
*b*c*d+187*b^2*c^2)/(d*x+c)^(17/6)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)

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Fricas [A]  time = 0.226384, size = 317, normalized size = 3.14 \[ \frac{6 \,{\left (72 \, b^{3} d^{2} x^{3} + 187 \, a b^{2} c^{2} - 170 \, a^{2} b c d + 55 \, a^{3} d^{2} + 12 \,{\left (17 \, b^{3} c d + a b^{2} d^{2}\right )} x^{2} +{\left (187 \, b^{3} c^{2} + 34 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x\right )}}{935 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} +{\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)),x, algorithm="fricas")

[Out]

6/935*(72*b^3*d^2*x^3 + 187*a*b^2*c^2 - 170*a^2*b*c*d + 55*a^3*d^2 + 12*(17*b^3*
c*d + a*b^2*d^2)*x^2 + (187*b^3*c^2 + 34*a*b^2*c*d - 5*a^2*b*d^2)*x)/((b^3*c^5 -
3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3 + (b^3*c^3*d^2 - 3*a*b^2*c^2*d^3
+ 3*a^2*b*c*d^4 - a^3*d^5)*x^2 + 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^
3 - a^3*c*d^4)*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)**(23/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(23/6)), x)