3.1711 \(\int \frac{1}{(a+b x)^{13/4} (c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{128 d^2 \sqrt [4]{c+d x}}{45 \sqrt [4]{a+b x} (b c-a d)^3}+\frac{32 d \sqrt [4]{c+d x}}{45 (a+b x)^{5/4} (b c-a d)^2}-\frac{4 \sqrt [4]{c+d x}}{9 (a+b x)^{9/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(1/4))/(9*(b*c - a*d)*(a + b*x)^(9/4)) + (32*d*(c + d*x)^(1/4))/(4
5*(b*c - a*d)^2*(a + b*x)^(5/4)) - (128*d^2*(c + d*x)^(1/4))/(45*(b*c - a*d)^3*(
a + b*x)^(1/4))

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Rubi [A]  time = 0.081931, 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 \[ -\frac{128 d^2 \sqrt [4]{c+d x}}{45 \sqrt [4]{a+b x} (b c-a d)^3}+\frac{32 d \sqrt [4]{c+d x}}{45 (a+b x)^{5/4} (b c-a d)^2}-\frac{4 \sqrt [4]{c+d x}}{9 (a+b x)^{9/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(13/4)*(c + d*x)^(3/4)),x]

[Out]

(-4*(c + d*x)^(1/4))/(9*(b*c - a*d)*(a + b*x)^(9/4)) + (32*d*(c + d*x)^(1/4))/(4
5*(b*c - a*d)^2*(a + b*x)^(5/4)) - (128*d^2*(c + d*x)^(1/4))/(45*(b*c - a*d)^3*(
a + b*x)^(1/4))

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Rubi in Sympy [A]  time = 12.1989, size = 88, normalized size = 0.87 \[ \frac{128 d^{2} \sqrt [4]{c + d x}}{45 \sqrt [4]{a + b x} \left (a d - b c\right )^{3}} + \frac{32 d \sqrt [4]{c + d x}}{45 \left (a + b x\right )^{\frac{5}{4}} \left (a d - b c\right )^{2}} + \frac{4 \sqrt [4]{c + d x}}{9 \left (a + b x\right )^{\frac{9}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(13/4)/(d*x+c)**(3/4),x)

[Out]

128*d**2*(c + d*x)**(1/4)/(45*(a + b*x)**(1/4)*(a*d - b*c)**3) + 32*d*(c + d*x)*
*(1/4)/(45*(a + b*x)**(5/4)*(a*d - b*c)**2) + 4*(c + d*x)**(1/4)/(9*(a + b*x)**(
9/4)*(a*d - b*c))

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Mathematica [A]  time = 0.0985909, size = 75, normalized size = 0.74 \[ -\frac{4 \sqrt [4]{c+d x} \left (45 a^2 d^2-18 a b d (c-4 d x)+b^2 \left (5 c^2-8 c d x+32 d^2 x^2\right )\right )}{45 (a+b x)^{9/4} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(13/4)*(c + d*x)^(3/4)),x]

[Out]

(-4*(c + d*x)^(1/4)*(45*a^2*d^2 - 18*a*b*d*(c - 4*d*x) + b^2*(5*c^2 - 8*c*d*x +
32*d^2*x^2)))/(45*(b*c - a*d)^3*(a + b*x)^(9/4))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[{\frac{128\,{b}^{2}{d}^{2}{x}^{2}+288\,ab{d}^{2}x-32\,{b}^{2}cdx+180\,{a}^{2}{d}^{2}-72\,abcd+20\,{b}^{2}{c}^{2}}{45\,{a}^{3}{d}^{3}-135\,{a}^{2}cb{d}^{2}+135\,a{b}^{2}{c}^{2}d-45\,{b}^{3}{c}^{3}}\sqrt [4]{dx+c} \left ( bx+a \right ) ^{-{\frac{9}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(13/4)/(d*x+c)^(3/4),x)

[Out]

4/45*(d*x+c)^(1/4)*(32*b^2*d^2*x^2+72*a*b*d^2*x-8*b^2*c*d*x+45*a^2*d^2-18*a*b*c*
d+5*b^2*c^2)/(b*x+a)^(9/4)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{13}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(13/4)*(d*x + c)^(3/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(13/4)*(d*x + c)^(3/4)), x)

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Fricas [A]  time = 0.213561, size = 339, normalized size = 3.36 \[ -\frac{4 \,{\left (32 \, b^{2} d^{2} x^{2} + 5 \, b^{2} c^{2} - 18 \, a b c d + 45 \, a^{2} d^{2} - 8 \,{\left (b^{2} c d - 9 \, a b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{45 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(13/4)*(d*x + c)^(3/4)),x, algorithm="fricas")

[Out]

-4/45*(32*b^2*d^2*x^2 + 5*b^2*c^2 - 18*a*b*c*d + 45*a^2*d^2 - 8*(b^2*c*d - 9*a*b
*d^2)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*
b*c*d^2 - a^6*d^3 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^
3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x^2 + 3*(a^2
*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*x)

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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)**(13/4)/(d*x+c)**(3/4),x)

[Out]

Timed out

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GIAC/XCAS [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)^(13/4)*(d*x + c)^(3/4)),x, algorithm="giac")

[Out]

Timed out