3.1735 \(\int \frac{1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{5};\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt{a+b x} \sqrt [5]{c+d x}} \]

[Out]

(-2*((b*(c + d*x))/(b*c - a*d))^(1/5)*Hypergeometric2F1[-1/2, 1/5, 1/2, -((d*(a
+ b*x))/(b*c - a*d))])/(b*Sqrt[a + b*x]*(c + d*x)^(1/5))

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Rubi [A]  time = 0.083009, antiderivative size = 72, 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 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{5};\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{b \sqrt{a+b x} \sqrt [5]{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(3/2)*(c + d*x)^(1/5)),x]

[Out]

(-2*((b*(c + d*x))/(b*c - a*d))^(1/5)*Hypergeometric2F1[-1/2, 1/5, 1/2, -((d*(a
+ b*x))/(b*c - a*d))])/(b*Sqrt[a + b*x]*(c + d*x)^(1/5))

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Rubi in Sympy [A]  time = 13.9482, size = 68, normalized size = 0.94 \[ \frac{5 d \sqrt{a + b x} \left (c + d x\right )^{\frac{4}{5}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{4}{5} \\ \frac{9}{5} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{4 \sqrt{\frac{d \left (a + b x\right )}{a d - b c}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/5),x)

[Out]

5*d*sqrt(a + b*x)*(c + d*x)**(4/5)*hyper((3/2, 4/5), (9/5,), b*(-c - d*x)/(a*d -
b*c))/(4*sqrt(d*(a + b*x)/(a*d - b*c))*(a*d - b*c)**2)

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Mathematica [A]  time = 0.139928, size = 84, normalized size = 1.17 \[ \frac{(c+d x)^{4/5} \left (3 \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{4}{5};\frac{9}{5};\frac{b (c+d x)}{b c-a d}\right )-8\right )}{4 \sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(1/5)),x]

[Out]

((c + d*x)^(4/5)*(-8 + 3*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]*Hypergeometric2F1[1/
2, 4/5, 9/5, (b*(c + d*x))/(b*c - a*d)]))/(4*(b*c - a*d)*Sqrt[a + b*x])

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [5]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(3/2)/(d*x+c)^(1/5),x)

[Out]

int(1/(b*x+a)^(3/2)/(d*x+c)^(1/5),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)),x, algorithm="fricas")

[Out]

integral(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \sqrt [5]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/5),x)

[Out]

Integral(1/((a + b*x)**(3/2)*(c + d*x)**(1/5)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(1/5)), x)