3.1786 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{35/6}} \, dx\)

Optimal. Leaf size=136 \[ \frac{7776 b^3 (a+b x)^{11/6}}{124729 (c+d x)^{11/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{11/6}}{11339 (c+d x)^{17/6} (b c-a d)^3}+\frac{108 b (a+b x)^{11/6}}{667 (c+d x)^{23/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(11/6))/(29*(b*c - a*d)*(c + d*x)^(29/6)) + (108*b*(a + b*x)^(11/6)
)/(667*(b*c - a*d)^2*(c + d*x)^(23/6)) + (1296*b^2*(a + b*x)^(11/6))/(11339*(b*c
- a*d)^3*(c + d*x)^(17/6)) + (7776*b^3*(a + b*x)^(11/6))/(124729*(b*c - a*d)^4*
(c + d*x)^(11/6))

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Rubi [A]  time = 0.119045, 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 \[ \frac{7776 b^3 (a+b x)^{11/6}}{124729 (c+d x)^{11/6} (b c-a d)^4}+\frac{1296 b^2 (a+b x)^{11/6}}{11339 (c+d x)^{17/6} (b c-a d)^3}+\frac{108 b (a+b x)^{11/6}}{667 (c+d x)^{23/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{29 (c+d x)^{29/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/6)/(c + d*x)^(35/6),x]

[Out]

(6*(a + b*x)^(11/6))/(29*(b*c - a*d)*(c + d*x)^(29/6)) + (108*b*(a + b*x)^(11/6)
)/(667*(b*c - a*d)^2*(c + d*x)^(23/6)) + (1296*b^2*(a + b*x)^(11/6))/(11339*(b*c
- a*d)^3*(c + d*x)^(17/6)) + (7776*b^3*(a + b*x)^(11/6))/(124729*(b*c - a*d)^4*
(c + d*x)^(11/6))

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Rubi in Sympy [A]  time = 19.6731, size = 121, normalized size = 0.89 \[ \frac{7776 b^{3} \left (a + b x\right )^{\frac{11}{6}}}{124729 \left (c + d x\right )^{\frac{11}{6}} \left (a d - b c\right )^{4}} - \frac{1296 b^{2} \left (a + b x\right )^{\frac{11}{6}}}{11339 \left (c + d x\right )^{\frac{17}{6}} \left (a d - b c\right )^{3}} + \frac{108 b \left (a + b x\right )^{\frac{11}{6}}}{667 \left (c + d x\right )^{\frac{23}{6}} \left (a d - b c\right )^{2}} - \frac{6 \left (a + b x\right )^{\frac{11}{6}}}{29 \left (c + d x\right )^{\frac{29}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/6)/(d*x+c)**(35/6),x)

[Out]

7776*b**3*(a + b*x)**(11/6)/(124729*(c + d*x)**(11/6)*(a*d - b*c)**4) - 1296*b**
2*(a + b*x)**(11/6)/(11339*(c + d*x)**(17/6)*(a*d - b*c)**3) + 108*b*(a + b*x)**
(11/6)/(667*(c + d*x)**(23/6)*(a*d - b*c)**2) - 6*(a + b*x)**(11/6)/(29*(c + d*x
)**(29/6)*(a*d - b*c))

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Mathematica [A]  time = 0.163194, size = 118, normalized size = 0.87 \[ \frac{6 (a+b x)^{11/6} \left (-4301 a^3 d^3+561 a^2 b d^2 (29 c+6 d x)-33 a b^2 d \left (667 c^2+348 c d x+72 d^2 x^2\right )+b^3 \left (11339 c^3+12006 c^2 d x+6264 c d^2 x^2+1296 d^3 x^3\right )\right )}{124729 (c+d x)^{29/6} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/6)/(c + d*x)^(35/6),x]

[Out]

(6*(a + b*x)^(11/6)*(-4301*a^3*d^3 + 561*a^2*b*d^2*(29*c + 6*d*x) - 33*a*b^2*d*(
667*c^2 + 348*c*d*x + 72*d^2*x^2) + b^3*(11339*c^3 + 12006*c^2*d*x + 6264*c*d^2*
x^2 + 1296*d^3*x^3)))/(124729*(b*c - a*d)^4*(c + d*x)^(29/6))

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Maple [A]  time = 0.013, size = 171, normalized size = 1.3 \[ -{\frac{-7776\,{x}^{3}{b}^{3}{d}^{3}+14256\,a{b}^{2}{d}^{3}{x}^{2}-37584\,{b}^{3}c{d}^{2}{x}^{2}-20196\,{a}^{2}b{d}^{3}x+68904\,a{b}^{2}c{d}^{2}x-72036\,{b}^{3}{c}^{2}dx+25806\,{a}^{3}{d}^{3}-97614\,{a}^{2}cb{d}^{2}+132066\,a{b}^{2}{c}^{2}d-68034\,{b}^{3}{c}^{3}}{124729\,{d}^{4}{a}^{4}-498916\,{a}^{3}bc{d}^{3}+748374\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-498916\,a{b}^{3}{c}^{3}d+124729\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{{\frac{11}{6}}} \left ( dx+c \right ) ^{-{\frac{29}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/6)/(d*x+c)^(35/6),x)

[Out]

-6/124729*(b*x+a)^(11/6)*(-1296*b^3*d^3*x^3+2376*a*b^2*d^3*x^2-6264*b^3*c*d^2*x^
2-3366*a^2*b*d^3*x+11484*a*b^2*c*d^2*x-12006*b^3*c^2*d*x+4301*a^3*d^3-16269*a^2*
b*c*d^2+22011*a*b^2*c^2*d-11339*b^3*c^3)/(d*x+c)^(29/6)/(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. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{35}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(35/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(35/6), x)

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Fricas [A]  time = 0.245005, size = 702, normalized size = 5.16 \[ \frac{6 \,{\left (1296 \, b^{5} d^{3} x^{5} + 11339 \, a^{2} b^{3} c^{3} - 22011 \, a^{3} b^{2} c^{2} d + 16269 \, a^{4} b c d^{2} - 4301 \, a^{5} d^{3} + 216 \,{\left (29 \, b^{5} c d^{2} + a b^{4} d^{3}\right )} x^{4} + 18 \,{\left (667 \, b^{5} c^{2} d + 58 \, a b^{4} c d^{2} - 5 \, a^{2} b^{3} d^{3}\right )} x^{3} +{\left (11339 \, b^{5} c^{3} + 2001 \, a b^{4} c^{2} d - 435 \, a^{2} b^{3} c d^{2} + 55 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (11339 \, a b^{4} c^{3} - 16008 \, a^{2} b^{3} c^{2} d + 10527 \, a^{3} b^{2} c d^{2} - 2618 \, a^{4} b d^{3}\right )} x\right )}}{124729 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4} +{\left (b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} x^{4} + 4 \,{\left (b^{4} c^{5} d^{3} - 4 \, a b^{3} c^{4} d^{4} + 6 \, a^{2} b^{2} c^{3} d^{5} - 4 \, a^{3} b c^{2} d^{6} + a^{4} c d^{7}\right )} x^{3} + 6 \,{\left (b^{4} c^{6} d^{2} - 4 \, a b^{3} c^{5} d^{3} + 6 \, a^{2} b^{2} c^{4} d^{4} - 4 \, a^{3} b c^{3} d^{5} + a^{4} c^{2} d^{6}\right )} x^{2} + 4 \,{\left (b^{4} c^{7} d - 4 \, a b^{3} c^{6} d^{2} + 6 \, a^{2} b^{2} c^{5} d^{3} - 4 \, a^{3} b c^{4} d^{4} + a^{4} c^{3} d^{5}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(35/6),x, algorithm="fricas")

[Out]

6/124729*(1296*b^5*d^3*x^5 + 11339*a^2*b^3*c^3 - 22011*a^3*b^2*c^2*d + 16269*a^4
*b*c*d^2 - 4301*a^5*d^3 + 216*(29*b^5*c*d^2 + a*b^4*d^3)*x^4 + 18*(667*b^5*c^2*d
+ 58*a*b^4*c*d^2 - 5*a^2*b^3*d^3)*x^3 + (11339*b^5*c^3 + 2001*a*b^4*c^2*d - 435
*a^2*b^3*c*d^2 + 55*a^3*b^2*d^3)*x^2 + 2*(11339*a*b^4*c^3 - 16008*a^2*b^3*c^2*d
+ 10527*a^3*b^2*c*d^2 - 2618*a^4*b*d^3)*x)/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2
*c^6*d^2 - 4*a^3*b*c^5*d^3 + a^4*c^4*d^4 + (b^4*c^4*d^4 - 4*a*b^3*c^3*d^5 + 6*a^
2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*x^4 + 4*(b^4*c^5*d^3 - 4*a*b^3*c^4*d^4
+ 6*a^2*b^2*c^3*d^5 - 4*a^3*b*c^2*d^6 + a^4*c*d^7)*x^3 + 6*(b^4*c^6*d^2 - 4*a*b^
3*c^5*d^3 + 6*a^2*b^2*c^4*d^4 - 4*a^3*b*c^3*d^5 + a^4*c^2*d^6)*x^2 + 4*(b^4*c^7*
d - 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^5*d^3 - 4*a^3*b*c^4*d^4 + a^4*c^3*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((b*x+a)**(5/6)/(d*x+c)**(35/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{35}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(35/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(35/6), x)