3.1698 \(\int \frac{1}{(a+b x)^{11/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=66 \[ \frac{16 d (c+d x)^{3/4}}{21 (a+b x)^{3/4} (b c-a d)^2}-\frac{4 (c+d x)^{3/4}}{7 (a+b x)^{7/4} (b c-a d)} \]

[Out]

(-4*(c + d*x)^(3/4))/(7*(b*c - a*d)*(a + b*x)^(7/4)) + (16*d*(c + d*x)^(3/4))/(2
1*(b*c - a*d)^2*(a + b*x)^(3/4))

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Rubi [A]  time = 0.0498358, 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 \[ \frac{16 d (c+d x)^{3/4}}{21 (a+b x)^{3/4} (b c-a d)^2}-\frac{4 (c+d x)^{3/4}}{7 (a+b x)^{7/4} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(c + d*x)^(3/4))/(7*(b*c - a*d)*(a + b*x)^(7/4)) + (16*d*(c + d*x)^(3/4))/(2
1*(b*c - a*d)^2*(a + b*x)^(3/4))

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Rubi in Sympy [A]  time = 6.82151, size = 56, normalized size = 0.85 \[ \frac{16 d \left (c + d x\right )^{\frac{3}{4}}}{21 \left (a + b x\right )^{\frac{3}{4}} \left (a d - b c\right )^{2}} + \frac{4 \left (c + d x\right )^{\frac{3}{4}}}{7 \left (a + b x\right )^{\frac{7}{4}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0688213, size = 46, normalized size = 0.7 \[ \frac{4 (c+d x)^{3/4} (7 a d-3 b c+4 b d x)}{21 (a+b x)^{7/4} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 54, normalized size = 0.8 \[{\frac{16\,bdx+28\,ad-12\,bc}{21\,{a}^{2}{d}^{2}-42\,abcd+21\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{3}{4}}} \left ( bx+a \right ) ^{-{\frac{7}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21484, size = 138, normalized size = 2.09 \[ \frac{4 \,{\left (4 \, b d^{2} x^{2} - 3 \, b c^{2} + 7 \, a c d +{\left (b c d + 7 \, a d^{2}\right )} x\right )}}{21 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )}{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/21*(4*b*d^2*x^2 - 3*b*c^2 + 7*a*c*d + (b*c*d + 7*a*d^2)*x)/((a*b^2*c^2 - 2*a^2
*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*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)**(1/4),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)^(1/4)),x, algorithm="giac")

[Out]

Timed out