3.1785 \(\int \frac{(a+b x)^{5/6}}{(c+d x)^{29/6}} \, dx\)

Optimal. Leaf size=101 \[ \frac{432 b^2 (a+b x)^{11/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac{72 b (a+b x)^{11/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)} \]

[Out]

(6*(a + b*x)^(11/6))/(23*(b*c - a*d)*(c + d*x)^(23/6)) + (72*b*(a + b*x)^(11/6))
/(391*(b*c - a*d)^2*(c + d*x)^(17/6)) + (432*b^2*(a + b*x)^(11/6))/(4301*(b*c -
a*d)^3*(c + d*x)^(11/6))

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Rubi [A]  time = 0.0847501, 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 \[ \frac{432 b^2 (a+b x)^{11/6}}{4301 (c+d x)^{11/6} (b c-a d)^3}+\frac{72 b (a+b x)^{11/6}}{391 (c+d x)^{17/6} (b c-a d)^2}+\frac{6 (a+b x)^{11/6}}{23 (c+d x)^{23/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/6)/(c + d*x)^(29/6),x]

[Out]

(6*(a + b*x)^(11/6))/(23*(b*c - a*d)*(c + d*x)^(23/6)) + (72*b*(a + b*x)^(11/6))
/(391*(b*c - a*d)^2*(c + d*x)^(17/6)) + (432*b^2*(a + b*x)^(11/6))/(4301*(b*c -
a*d)^3*(c + d*x)^(11/6))

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Rubi in Sympy [A]  time = 12.5119, size = 88, normalized size = 0.87 \[ - \frac{432 b^{2} \left (a + b x\right )^{\frac{11}{6}}}{4301 \left (c + d x\right )^{\frac{11}{6}} \left (a d - b c\right )^{3}} + \frac{72 b \left (a + b x\right )^{\frac{11}{6}}}{391 \left (c + d x\right )^{\frac{17}{6}} \left (a d - b c\right )^{2}} - \frac{6 \left (a + b x\right )^{\frac{11}{6}}}{23 \left (c + d x\right )^{\frac{23}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/6)/(d*x+c)**(29/6),x)

[Out]

-432*b**2*(a + b*x)**(11/6)/(4301*(c + d*x)**(11/6)*(a*d - b*c)**3) + 72*b*(a +
b*x)**(11/6)/(391*(c + d*x)**(17/6)*(a*d - b*c)**2) - 6*(a + b*x)**(11/6)/(23*(c
+ d*x)**(23/6)*(a*d - b*c))

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Mathematica [A]  time = 0.106167, size = 77, normalized size = 0.76 \[ \frac{6 (a+b x)^{11/6} \left (187 a^2 d^2-22 a b d (23 c+6 d x)+b^2 \left (391 c^2+276 c d x+72 d^2 x^2\right )\right )}{4301 (c+d x)^{23/6} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/6)/(c + d*x)^(29/6),x]

[Out]

(6*(a + b*x)^(11/6)*(187*a^2*d^2 - 22*a*b*d*(23*c + 6*d*x) + b^2*(391*c^2 + 276*
c*d*x + 72*d^2*x^2)))/(4301*(b*c - a*d)^3*(c + d*x)^(23/6))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[ -{\frac{432\,{b}^{2}{d}^{2}{x}^{2}-792\,ab{d}^{2}x+1656\,{b}^{2}cdx+1122\,{a}^{2}{d}^{2}-3036\,abcd+2346\,{b}^{2}{c}^{2}}{4301\,{a}^{3}{d}^{3}-12903\,{a}^{2}cb{d}^{2}+12903\,a{b}^{2}{c}^{2}d-4301\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{{\frac{11}{6}}} \left ( dx+c \right ) ^{-{\frac{23}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/6)/(d*x+c)^(29/6),x)

[Out]

-6/4301*(b*x+a)^(11/6)*(72*b^2*d^2*x^2-132*a*b*d^2*x+276*b^2*c*d*x+187*a^2*d^2-5
06*a*b*c*d+391*b^2*c^2)/(d*x+c)^(23/6)/(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. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{29}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(29/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(29/6), x)

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Fricas [A]  time = 0.238176, size = 437, normalized size = 4.33 \[ \frac{6 \,{\left (72 \, b^{4} d^{2} x^{4} + 391 \, a^{2} b^{2} c^{2} - 506 \, a^{3} b c d + 187 \, a^{4} d^{2} + 12 \,{\left (23 \, b^{4} c d + a b^{3} d^{2}\right )} x^{3} +{\left (391 \, b^{4} c^{2} + 46 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (391 \, a b^{3} c^{2} - 368 \, a^{2} b^{2} c d + 121 \, a^{3} b d^{2}\right )} x\right )}}{4301 \,{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3} +{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{3} + 3 \,{\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2} + 3 \,{\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(29/6),x, algorithm="fricas")

[Out]

6/4301*(72*b^4*d^2*x^4 + 391*a^2*b^2*c^2 - 506*a^3*b*c*d + 187*a^4*d^2 + 12*(23*
b^4*c*d + a*b^3*d^2)*x^3 + (391*b^4*c^2 + 46*a*b^3*c*d - 5*a^2*b^2*d^2)*x^2 + 2*
(391*a*b^3*c^2 - 368*a^2*b^2*c*d + 121*a^3*b*d^2)*x)/((b^3*c^6 - 3*a*b^2*c^5*d +
3*a^2*b*c^4*d^2 - a^3*c^3*d^3 + (b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5
- a^3*d^6)*x^3 + 3*(b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)
*x^2 + 3*(b^3*c^5*d - 3*a*b^2*c^4*d^2 + 3*a^2*b*c^3*d^3 - a^3*c^2*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((b*x+a)**(5/6)/(d*x+c)**(29/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{5}{6}}}{{\left (d x + c\right )}^{\frac{29}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/6)/(d*x + c)^(29/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/6)/(d*x + c)^(29/6), x)