Optimal. Leaf size=63 \[ \frac{2 \pi ^n (b x)^{7/2} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]
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Rubi [A] time = 0.0936117, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 \pi ^n (b x)^{7/2} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]
Antiderivative was successfully verified.
[In] Int[(b*x)^(5/2)*(Pi + d*x)^n*(e + f*x)^p,x]
[Out]
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Rubi in Sympy [A] time = 11.4303, size = 49, normalized size = 0.78 \[ \frac{2 \pi ^{n} \left (b x\right )^{\frac{7}{2}} \left (1 + \frac{f x}{e}\right )^{- p} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (\frac{7}{2},- n,- p,\frac{9}{2},- \frac{d x}{\pi },- \frac{f x}{e} \right )}}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x)**(5/2)*(d*x+pi)**n*(f*x+e)**p,x)
[Out]
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Mathematica [B] time = 0.377245, size = 159, normalized size = 2.52 \[ \frac{18 \pi e x (b x)^{5/2} (d x+\pi )^n (e+f x)^p F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 \left (9 \pi e F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )+2 x \left (d e n F_1\left (\frac{9}{2};1-n,-p;\frac{11}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )+\pi f p F_1\left (\frac{9}{2};-n,1-p;\frac{11}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(b*x)^(5/2)*(Pi + d*x)^n*(e + f*x)^p,x]
[Out]
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Maple [F] time = 0.057, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{{\frac{5}{2}}} \left ( dx+\pi \right ) ^{n} \left ( fx+e \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x)^(5/2)*(d*x+Pi)^n*(f*x+e)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + e)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p} b^{2} x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^(5/2)*(pi + d*x)^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)**(5/2)*(d*x+pi)**n*(f*x+e)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + e)^p,x, algorithm="giac")
[Out]