Optimal. Leaf size=74 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{3}{2},\frac{1}{5};-\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}} \]
[Out]
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Rubi [A] time = 0.084057, antiderivative size = 74, 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 \[ -\frac{2 \sqrt [5]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{3}{2},\frac{1}{5};-\frac{1}{2};-\frac{d (a+b x)}{b c-a d}\right )}{3 b (a+b x)^{3/2} \sqrt [5]{c+d x}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(5/2)*(c + d*x)^(1/5)),x]
[Out]
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Rubi in Sympy [A] time = 14.5676, size = 70, normalized size = 0.95 \[ \frac{5 d^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{4}{5}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, \frac{4}{5} \\ \frac{9}{5} \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{4 \sqrt{\frac{d \left (a + b x\right )}{a d - b c}} \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/5),x)
[Out]
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Mathematica [A] time = 0.279729, size = 105, normalized size = 1.42 \[ \frac{(c+d x)^{4/5} \left (8 (12 a d-5 b c+7 b d x)-21 d (a+b x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{4}{5};\frac{9}{5};\frac{b (c+d x)}{b c-a d}\right )\right )}{60 (a+b x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(1/5)),x]
[Out]
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Maple [F] time = 0.072, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt [5]{dx+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(5/2)/(d*x+c)^(1/5),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/5)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{5}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/5)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \sqrt [5]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/5),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}}{\left (d x + c\right )}^{\frac{1}{5}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/5)),x, algorithm="giac")
[Out]