Optimal. Leaf size=79 \[ \frac{2 (b x)^{3/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{3 b} \]
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Rubi [A] time = 0.132804, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 (b x)^{3/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{3 b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x]*(c + d*x)^n*(e + f*x)^p,x]
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Rubi in Sympy [A] time = 18.3858, size = 61, normalized size = 0.77 \[ \frac{2 \left (b x\right )^{\frac{3}{2}} \left (1 + \frac{d x}{c}\right )^{- n} \left (1 + \frac{f x}{e}\right )^{- p} \left (c + d x\right )^{n} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (\frac{3}{2},- n,- p,\frac{5}{2},- \frac{d x}{c},- \frac{f x}{e} \right )}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x)**(1/2)*(d*x+c)**n*(f*x+e)**p,x)
[Out]
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Mathematica [A] time = 0.374932, size = 157, normalized size = 1.99 \[ \frac{10 c e x \sqrt{b x} (c+d x)^n (e+f x)^p F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{15 c e F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+6 d e n x F_1\left (\frac{5}{2};1-n,-p;\frac{7}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+6 c f p x F_1\left (\frac{5}{2};-n,1-p;\frac{7}{2};-\frac{d x}{c},-\frac{f x}{e}\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[b*x]*(c + d*x)^n*(e + f*x)^p,x]
[Out]
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Maple [F] time = 0.059, size = 0, normalized size = 0. \[ \int \sqrt{bx} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x)^(1/2)*(d*x+c)^n*(f*x+e)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x)*(d*x + c)^n*(f*x + e)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x)*(d*x + c)^n*(f*x + e)^p,x, algorithm="fricas")
[Out]
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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((b*x)**(1/2)*(d*x+c)**n*(f*x+e)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x)*(d*x + c)^n*(f*x + e)^p,x, algorithm="giac")
[Out]