3.1710 \(\int \frac{1}{(a+b x)^{9/4} (c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=66 \[ \frac{16 d \sqrt [4]{c+d x}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac{4 \sqrt [4]{c+d x}}{5 (a+b x)^{5/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(1/4))/(5*(b*c - a*d)*(a + b*x)^(5/4)) + (16*d*(c + d*x)^(1/4))/(5
*(b*c - a*d)^2*(a + b*x)^(1/4))

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Rubi [A]  time = 0.0505276, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 d \sqrt [4]{c+d x}}{5 \sqrt [4]{a+b x} (b c-a d)^2}-\frac{4 \sqrt [4]{c+d x}}{5 (a+b x)^{5/4} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(9/4)*(c + d*x)^(3/4)),x]

[Out]

(-4*(c + d*x)^(1/4))/(5*(b*c - a*d)*(a + b*x)^(5/4)) + (16*d*(c + d*x)^(1/4))/(5
*(b*c - a*d)^2*(a + b*x)^(1/4))

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Rubi in Sympy [A]  time = 6.78994, size = 56, normalized size = 0.85 \[ \frac{16 d \sqrt [4]{c + d x}}{5 \sqrt [4]{a + b x} \left (a d - b c\right )^{2}} + \frac{4 \sqrt [4]{c + d x}}{5 \left (a + b x\right )^{\frac{5}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(9/4)/(d*x+c)**(3/4),x)

[Out]

16*d*(c + d*x)**(1/4)/(5*(a + b*x)**(1/4)*(a*d - b*c)**2) + 4*(c + d*x)**(1/4)/(
5*(a + b*x)**(5/4)*(a*d - b*c))

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Mathematica [A]  time = 0.0628399, size = 46, normalized size = 0.7 \[ \frac{4 \sqrt [4]{c+d x} (5 a d-b c+4 b d x)}{5 (a+b x)^{5/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(9/4)*(c + d*x)^(3/4)),x]

[Out]

(4*(c + d*x)^(1/4)*(-(b*c) + 5*a*d + 4*b*d*x))/(5*(b*c - a*d)^2*(a + b*x)^(5/4))

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Maple [A]  time = 0.009, size = 54, normalized size = 0.8 \[{\frac{16\,bdx+20\,ad-4\,bc}{5\,{a}^{2}{d}^{2}-10\,abcd+5\,{b}^{2}{c}^{2}}\sqrt [4]{dx+c} \left ( bx+a \right ) ^{-{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(9/4)/(d*x+c)^(3/4),x)

[Out]

4/5*(d*x+c)^(1/4)*(4*b*d*x+5*a*d-b*c)/(b*x+a)^(5/4)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{9}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/4)*(d*x + c)^(3/4)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(9/4)*(d*x + c)^(3/4)), x)

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Fricas [A]  time = 0.225615, size = 159, normalized size = 2.41 \[ \frac{4 \,{\left (4 \, b d x - b c + 5 \, a d\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}{5 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(9/4)*(d*x + c)^(3/4)),x, algorithm="fricas")

[Out]

4/5*(4*b*d*x - b*c + 5*a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4)/(a^2*b^2*c^2 - 2*a^3
*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 2*(a*b^3*c^2 - 2*
a^2*b^2*c*d + a^3*b*d^2)*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)**(9/4)/(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)^(9/4)*(d*x + c)^(3/4)),x, algorithm="giac")

[Out]

Timed out