3.1838 \(\int \frac{1}{(a+b x)^{7/6} (c+d x)^{23/6}} \, dx\)

Optimal. Leaf size=134 \[ -\frac{7776 b^2 d (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^4}-\frac{1296 b d (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^3}-\frac{108 d (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)} \]

[Out]

-6/((b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(17/6)) - (108*d*(a + b*x)^(5/6))/(17*
(b*c - a*d)^2*(c + d*x)^(17/6)) - (1296*b*d*(a + b*x)^(5/6))/(187*(b*c - a*d)^3*
(c + d*x)^(11/6)) - (7776*b^2*d*(a + b*x)^(5/6))/(935*(b*c - a*d)^4*(c + d*x)^(5
/6))

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Rubi [A]  time = 0.124579, antiderivative size = 134, 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 \[ -\frac{7776 b^2 d (a+b x)^{5/6}}{935 (c+d x)^{5/6} (b c-a d)^4}-\frac{1296 b d (a+b x)^{5/6}}{187 (c+d x)^{11/6} (b c-a d)^3}-\frac{108 d (a+b x)^{5/6}}{17 (c+d x)^{17/6} (b c-a d)^2}-\frac{6}{\sqrt [6]{a+b x} (c+d x)^{17/6} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(7/6)*(c + d*x)^(23/6)),x]

[Out]

-6/((b*c - a*d)*(a + b*x)^(1/6)*(c + d*x)^(17/6)) - (108*d*(a + b*x)^(5/6))/(17*
(b*c - a*d)^2*(c + d*x)^(17/6)) - (1296*b*d*(a + b*x)^(5/6))/(187*(b*c - a*d)^3*
(c + d*x)^(11/6)) - (7776*b^2*d*(a + b*x)^(5/6))/(935*(b*c - a*d)^4*(c + d*x)^(5
/6))

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Rubi in Sympy [A]  time = 19.3749, size = 121, normalized size = 0.9 \[ - \frac{7776 b^{2} d \left (a + b x\right )^{\frac{5}{6}}}{935 \left (c + d x\right )^{\frac{5}{6}} \left (a d - b c\right )^{4}} + \frac{1296 b d \left (a + b x\right )^{\frac{5}{6}}}{187 \left (c + d x\right )^{\frac{11}{6}} \left (a d - b c\right )^{3}} - \frac{108 d \left (a + b x\right )^{\frac{5}{6}}}{17 \left (c + d x\right )^{\frac{17}{6}} \left (a d - b c\right )^{2}} + \frac{6}{\sqrt [6]{a + b x} \left (c + d x\right )^{\frac{17}{6}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(7/6)/(d*x+c)**(23/6),x)

[Out]

-7776*b**2*d*(a + b*x)**(5/6)/(935*(c + d*x)**(5/6)*(a*d - b*c)**4) + 1296*b*d*(
a + b*x)**(5/6)/(187*(c + d*x)**(11/6)*(a*d - b*c)**3) - 108*d*(a + b*x)**(5/6)/
(17*(c + d*x)**(17/6)*(a*d - b*c)**2) + 6/((a + b*x)**(1/6)*(c + d*x)**(17/6)*(a
*d - b*c))

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Mathematica [A]  time = 0.234569, size = 97, normalized size = 0.72 \[ \frac{6 (a+b x)^{5/6} \sqrt [6]{c+d x} \left (-\frac{935 b^3}{a+b x}-\frac{145 b d (b c-a d)}{(c+d x)^2}-\frac{55 d (b c-a d)^2}{(c+d x)^3}-\frac{361 b^2 d}{c+d x}\right )}{935 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(7/6)*(c + d*x)^(23/6)),x]

[Out]

(6*(a + b*x)^(5/6)*(c + d*x)^(1/6)*((-935*b^3)/(a + b*x) - (55*d*(b*c - a*d)^2)/
(c + d*x)^3 - (145*b*d*(b*c - a*d))/(c + d*x)^2 - (361*b^2*d)/(c + d*x)))/(935*(
b*c - a*d)^4)

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Maple [A]  time = 0.013, size = 171, normalized size = 1.3 \[ -{\frac{7776\,{x}^{3}{b}^{3}{d}^{3}+1296\,a{b}^{2}{d}^{3}{x}^{2}+22032\,{b}^{3}c{d}^{2}{x}^{2}-540\,{a}^{2}b{d}^{3}x+3672\,a{b}^{2}c{d}^{2}x+20196\,{b}^{3}{c}^{2}dx+330\,{a}^{3}{d}^{3}-1530\,{a}^{2}cb{d}^{2}+3366\,a{b}^{2}{c}^{2}d+5610\,{b}^{3}{c}^{3}}{935\,{a}^{4}{d}^{4}-3740\,{a}^{3}bc{d}^{3}+5610\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-3740\,a{b}^{3}{c}^{3}d+935\,{b}^{4}{c}^{4}}{\frac{1}{\sqrt [6]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{17}{6}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(7/6)/(d*x+c)^(23/6),x)

[Out]

-6/935*(1296*b^3*d^3*x^3+216*a*b^2*d^3*x^2+3672*b^3*c*d^2*x^2-90*a^2*b*d^3*x+612
*a*b^2*c*d^2*x+3366*b^3*c^2*d*x+55*a^3*d^3-255*a^2*b*c*d^2+561*a*b^2*c^2*d+935*b
^3*c^3)/(b*x+a)^(1/6)/(d*x+c)^(17/6)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(23/6)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(23/6)), x)

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Fricas [A]  time = 0.222684, size = 393, normalized size = 2.93 \[ -\frac{6 \,{\left (1296 \, b^{3} d^{3} x^{3} + 935 \, b^{3} c^{3} + 561 \, a b^{2} c^{2} d - 255 \, a^{2} b c d^{2} + 55 \, a^{3} d^{3} + 216 \,{\left (17 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 18 \,{\left (187 \, b^{3} c^{2} d + 34 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x\right )}}{935 \,{\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4} +{\left (b^{4} c^{4} d^{2} - 4 \, a b^{3} c^{3} d^{3} + 6 \, a^{2} b^{2} c^{2} d^{4} - 4 \, a^{3} b c d^{5} + a^{4} d^{6}\right )} x^{2} + 2 \,{\left (b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}\right )} x\right )}{\left (b x + a\right )}^{\frac{1}{6}}{\left (d x + c\right )}^{\frac{5}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(23/6)),x, algorithm="fricas")

[Out]

-6/935*(1296*b^3*d^3*x^3 + 935*b^3*c^3 + 561*a*b^2*c^2*d - 255*a^2*b*c*d^2 + 55*
a^3*d^3 + 216*(17*b^3*c*d^2 + a*b^2*d^3)*x^2 + 18*(187*b^3*c^2*d + 34*a*b^2*c*d^
2 - 5*a^2*b*d^3)*x)/((b^4*c^6 - 4*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*
d^3 + a^4*c^2*d^4 + (b^4*c^4*d^2 - 4*a*b^3*c^3*d^3 + 6*a^2*b^2*c^2*d^4 - 4*a^3*b
*c*d^5 + a^4*d^6)*x^2 + 2*(b^4*c^5*d - 4*a*b^3*c^4*d^2 + 6*a^2*b^2*c^3*d^3 - 4*a
^3*b*c^2*d^4 + a^4*c*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)**(7/6)/(d*x+c)**(23/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{7}{6}}{\left (d x + c\right )}^{\frac{23}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(7/6)*(d*x + c)^(23/6)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(7/6)*(d*x + c)^(23/6)), x)