3.1812 \(\int \frac{1}{\sqrt [6]{a+b x} (c+d x)^{29/6}} \, dx\)

Optimal. Leaf size=136 \[ \frac{7776 b^3 (a+b x)^{5/6}}{21505 (c+d x)^{5/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{5/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac{108 b (a+b x)^{5/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{23 (c+d x)^{23/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(5/6))/(23*(b*c - a*d)*(c + d*x)^(23/6)) + (108*b*(a + b*x)^(5/6))/
(391*(b*c - a*d)^2*(c + d*x)^(17/6)) + (1296*b^2*(a + b*x)^(5/6))/(4301*(b*c - a
*d)^3*(c + d*x)^(11/6)) + (7776*b^3*(a + b*x)^(5/6))/(21505*(b*c - a*d)^4*(c + d
*x)^(5/6))

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Rubi [A]  time = 0.120106, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{7776 b^3 (a+b x)^{5/6}}{21505 (c+d x)^{5/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{5/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac{108 b (a+b x)^{5/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac{6 (a+b x)^{5/6}}{23 (c+d x)^{23/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(1/6)*(c + d*x)^(29/6)),x]

[Out]

(6*(a + b*x)^(5/6))/(23*(b*c - a*d)*(c + d*x)^(23/6)) + (108*b*(a + b*x)^(5/6))/
(391*(b*c - a*d)^2*(c + d*x)^(17/6)) + (1296*b^2*(a + b*x)^(5/6))/(4301*(b*c - a
*d)^3*(c + d*x)^(11/6)) + (7776*b^3*(a + b*x)^(5/6))/(21505*(b*c - a*d)^4*(c + d
*x)^(5/6))

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Rubi in Sympy [A]  time = 19.9706, size = 121, normalized size = 0.89 \[ \frac{7776 b^{3} \left (a + b x\right )^{\frac{5}{6}}}{21505 \left (c + d x\right )^{\frac{5}{6}} \left (a d - b c\right )^{4}} - \frac{1296 b^{2} \left (a + b x\right )^{\frac{5}{6}}}{4301 \left (c + d x\right )^{\frac{11}{6}} \left (a d - b c\right )^{3}} + \frac{108 b \left (a + b x\right )^{\frac{5}{6}}}{391 \left (c + d x\right )^{\frac{17}{6}} \left (a d - b c\right )^{2}} - \frac{6 \left (a + b x\right )^{\frac{5}{6}}}{23 \left (c + d x\right )^{\frac{23}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/6)/(d*x+c)**(29/6),x)

[Out]

7776*b**3*(a + b*x)**(5/6)/(21505*(c + d*x)**(5/6)*(a*d - b*c)**4) - 1296*b**2*(
a + b*x)**(5/6)/(4301*(c + d*x)**(11/6)*(a*d - b*c)**3) + 108*b*(a + b*x)**(5/6)
/(391*(c + d*x)**(17/6)*(a*d - b*c)**2) - 6*(a + b*x)**(5/6)/(23*(c + d*x)**(23/
6)*(a*d - b*c))

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Mathematica [A]  time = 0.189617, size = 95, normalized size = 0.7 \[ \frac{6 (a+b x)^{5/6} \left (1080 b^2 (c+d x)^2 (b c-a d)+990 b (c+d x) (b c-a d)^2+935 (b c-a d)^3+1296 b^3 (c+d x)^3\right )}{21505 (c+d x)^{23/6} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(1/6)*(c + d*x)^(29/6)),x]

[Out]

(6*(a + b*x)^(5/6)*(935*(b*c - a*d)^3 + 990*b*(b*c - a*d)^2*(c + d*x) + 1080*b^2
*(b*c - a*d)*(c + d*x)^2 + 1296*b^3*(c + d*x)^3))/(21505*(b*c - a*d)^4*(c + d*x)
^(23/6))

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Maple [A]  time = 0.013, size = 171, normalized size = 1.3 \[ -{\frac{-7776\,{x}^{3}{b}^{3}{d}^{3}+6480\,a{b}^{2}{d}^{3}{x}^{2}-29808\,{b}^{3}c{d}^{2}{x}^{2}-5940\,{a}^{2}b{d}^{3}x+24840\,a{b}^{2}c{d}^{2}x-42228\,{b}^{3}{c}^{2}dx+5610\,{a}^{3}{d}^{3}-22770\,{a}^{2}cb{d}^{2}+35190\,a{b}^{2}{c}^{2}d-25806\,{b}^{3}{c}^{3}}{21505\,{a}^{4}{d}^{4}-86020\,{a}^{3}bc{d}^{3}+129030\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-86020\,a{b}^{3}{c}^{3}d+21505\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{{\frac{5}{6}}} \left ( dx+c \right ) ^{-{\frac{23}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/6)/(d*x+c)^(29/6),x)

[Out]

-6/21505*(b*x+a)^(5/6)*(-1296*b^3*d^3*x^3+1080*a*b^2*d^3*x^2-4968*b^3*c*d^2*x^2-
990*a^2*b*d^3*x+4140*a*b^2*c*d^2*x-7038*b^3*c^2*d*x+935*a^3*d^3-3795*a^2*b*c*d^2
+5865*a*b^2*c^2*d-4301*b^3*c^3)/(d*x+c)^(23/6)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*
c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{29}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(29/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(29/6)), x)

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Fricas [A]  time = 0.23598, size = 545, normalized size = 4.01 \[ \frac{6 \,{\left (1296 \, b^{4} d^{3} x^{4} + 4301 \, a b^{3} c^{3} - 5865 \, a^{2} b^{2} c^{2} d + 3795 \, a^{3} b c d^{2} - 935 \, a^{4} d^{3} + 216 \,{\left (23 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} x^{3} + 18 \,{\left (391 \, b^{4} c^{2} d + 46 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{2} +{\left (4301 \, b^{4} c^{3} + 1173 \, a b^{3} c^{2} d - 345 \, a^{2} b^{2} c d^{2} + 55 \, a^{3} b d^{3}\right )} x\right )}}{21505 \,{\left (b^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4} +{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} x^{3} + 3 \,{\left (b^{4} c^{5} d^{2} - 4 \, a b^{3} c^{4} d^{3} + 6 \, a^{2} b^{2} c^{3} d^{4} - 4 \, a^{3} b c^{2} d^{5} + a^{4} c d^{6}\right )} x^{2} + 3 \,{\left (b^{4} c^{6} d - 4 \, a b^{3} c^{5} d^{2} + 6 \, a^{2} b^{2} c^{4} d^{3} - 4 \, a^{3} b c^{3} d^{4} + a^{4} c^{2} d^{5}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(29/6)),x, algorithm="fricas")

[Out]

6/21505*(1296*b^4*d^3*x^4 + 4301*a*b^3*c^3 - 5865*a^2*b^2*c^2*d + 3795*a^3*b*c*d
^2 - 935*a^4*d^3 + 216*(23*b^4*c*d^2 + a*b^3*d^3)*x^3 + 18*(391*b^4*c^2*d + 46*a
*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + (4301*b^4*c^3 + 1173*a*b^3*c^2*d - 345*a^2*b^2
*c*d^2 + 55*a^3*b*d^3)*x)/((b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*
b*c^4*d^3 + a^4*c^3*d^4 + (b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4
*a^3*b*c*d^6 + a^4*d^7)*x^3 + 3*(b^4*c^5*d^2 - 4*a*b^3*c^4*d^3 + 6*a^2*b^2*c^3*d
^4 - 4*a^3*b*c^2*d^5 + a^4*c*d^6)*x^2 + 3*(b^4*c^6*d - 4*a*b^3*c^5*d^2 + 6*a^2*b
^2*c^4*d^3 - 4*a^3*b*c^3*d^4 + a^4*c^2*d^5)*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)**(29/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{29}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/6)*(d*x + c)^(29/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/6)*(d*x + c)^(29/6)), x)