3.1722 \(\int \frac{1}{(a+b x)^{11/4} (c+d x)^{5/4}} \, dx\)

Optimal. Leaf size=101 \[ \frac{128 d^2 \sqrt [4]{a+b x}}{21 \sqrt [4]{c+d x} (b c-a d)^3}+\frac{32 d}{21 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}-\frac{4}{7 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)} \]

[Out]

-4/(7*(b*c - a*d)*(a + b*x)^(7/4)*(c + d*x)^(1/4)) + (32*d)/(21*(b*c - a*d)^2*(a
+ b*x)^(3/4)*(c + d*x)^(1/4)) + (128*d^2*(a + b*x)^(1/4))/(21*(b*c - a*d)^3*(c
+ d*x)^(1/4))

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Rubi [A]  time = 0.0816494, 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 \[ \frac{128 d^2 \sqrt [4]{a+b x}}{21 \sqrt [4]{c+d x} (b c-a d)^3}+\frac{32 d}{21 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}-\frac{4}{7 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(11/4)*(c + d*x)^(5/4)),x]

[Out]

-4/(7*(b*c - a*d)*(a + b*x)^(7/4)*(c + d*x)^(1/4)) + (32*d)/(21*(b*c - a*d)^2*(a
+ b*x)^(3/4)*(c + d*x)^(1/4)) + (128*d^2*(a + b*x)^(1/4))/(21*(b*c - a*d)^3*(c
+ d*x)^(1/4))

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Rubi in Sympy [A]  time = 12.264, size = 88, normalized size = 0.87 \[ - \frac{128 d^{2} \sqrt [4]{a + b x}}{21 \sqrt [4]{c + d x} \left (a d - b c\right )^{3}} + \frac{32 d}{21 \left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x} \left (a d - b c\right )^{2}} + \frac{4}{7 \left (a + b x\right )^{\frac{7}{4}} \sqrt [4]{c + d x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(11/4)/(d*x+c)**(5/4),x)

[Out]

-128*d**2*(a + b*x)**(1/4)/(21*(c + d*x)**(1/4)*(a*d - b*c)**3) + 32*d/(21*(a +
b*x)**(3/4)*(c + d*x)**(1/4)*(a*d - b*c)**2) + 4/(7*(a + b*x)**(7/4)*(c + d*x)**
(1/4)*(a*d - b*c))

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Mathematica [A]  time = 0.118972, size = 76, normalized size = 0.75 \[ \frac{84 a^2 d^2+56 a b d (c+4 d x)+4 b^2 \left (-3 c^2+8 c d x+32 d^2 x^2\right )}{21 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(11/4)*(c + d*x)^(5/4)),x]

[Out]

(84*a^2*d^2 + 56*a*b*d*(c + 4*d*x) + 4*b^2*(-3*c^2 + 8*c*d*x + 32*d^2*x^2))/(21*
(b*c - a*d)^3*(a + b*x)^(7/4)*(c + d*x)^(1/4))

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Maple [A]  time = 0.01, size = 105, normalized size = 1. \[ -{\frac{128\,{b}^{2}{d}^{2}{x}^{2}+224\,ab{d}^{2}x+32\,{b}^{2}cdx+84\,{a}^{2}{d}^{2}+56\,abcd-12\,{b}^{2}{c}^{2}}{21\,{a}^{3}{d}^{3}-63\,{a}^{2}cb{d}^{2}+63\,a{b}^{2}{c}^{2}d-21\,{b}^{3}{c}^{3}} \left ( bx+a \right ) ^{-{\frac{7}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(11/4)/(d*x+c)^(5/4),x)

[Out]

-4/21*(32*b^2*d^2*x^2+56*a*b*d^2*x+8*b^2*c*d*x+21*a^2*d^2+14*a*b*c*d-3*b^2*c^2)/
(b*x+a)^(7/4)/(d*x+c)^(1/4)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{11}{4}}{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/4)*(d*x + c)^(5/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(11/4)*(d*x + c)^(5/4)), x)

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Fricas [A]  time = 0.216311, size = 201, normalized size = 1.99 \[ \frac{4 \,{\left (32 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 14 \, a b c d + 21 \, a^{2} d^{2} + 8 \,{\left (b^{2} c d + 7 \, a b d^{2}\right )} x\right )}}{21 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(11/4)*(d*x + c)^(5/4)),x, algorithm="fricas")

[Out]

4/21*(32*b^2*d^2*x^2 - 3*b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2 + 8*(b^2*c*d + 7*a*b*
d^2)*x)/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3 + (b^4*c^3 - 3*a
*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4))

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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)**(11/4)/(d*x+c)**(5/4),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)^(11/4)*(d*x + c)^(5/4)),x, algorithm="giac")

[Out]

Timed out