3.938 \(\int \frac{(b x)^{5/2} (c+d x)^n}{e+f x} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 (b x)^{7/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (\frac{7}{2};-n,1;\frac{9}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{7 b e} \]

[Out]

(2*(b*x)^(7/2)*(c + d*x)^n*AppellF1[7/2, -n, 1, 9/2, -((d*x)/c), -((f*x)/e)])/(7
*b*e*(1 + (d*x)/c)^n)

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Rubi [A]  time = 0.219879, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 (b x)^{7/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (\frac{7}{2};-n,1;\frac{9}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{7 b e} \]

Antiderivative was successfully verified.

[In]  Int[((b*x)^(5/2)*(c + d*x)^n)/(e + f*x),x]

[Out]

(2*(b*x)^(7/2)*(c + d*x)^n*AppellF1[7/2, -n, 1, 9/2, -((d*x)/c), -((f*x)/e)])/(7
*b*e*(1 + (d*x)/c)^n)

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Rubi in Sympy [A]  time = 11.3662, size = 46, normalized size = 0.75 \[ \frac{2 \left (b x\right )^{\frac{7}{2}} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (\frac{7}{2},1,- n,\frac{9}{2},- \frac{f x}{e},- \frac{d x}{c} \right )}}{7 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x)**(5/2)*(d*x+c)**n/(f*x+e),x)

[Out]

2*(b*x)**(7/2)*(1 + d*x/c)**(-n)*(c + d*x)**n*appellf1(7/2, 1, -n, 9/2, -f*x/e,
-d*x/c)/(7*b*e)

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Mathematica [B]  time = 0.72477, size = 239, normalized size = 3.92 \[ \frac{2 (b x)^{5/2} (c+d x)^n \left (\left (\frac{d x}{c}+1\right )^{-n} \left (15 e^2 \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{d x}{c}\right )+f x \left (3 f x \, _2F_1\left (\frac{5}{2},-n;\frac{7}{2};-\frac{d x}{c}\right )-5 e \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{d x}{c}\right )\right )\right )-\frac{45 c e^4 F_1\left (\frac{1}{2};-n,1;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{(e+f x) \left (3 c e F_1\left (\frac{1}{2};-n,1;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+2 d e n x F_1\left (\frac{3}{2};1-n,1;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )-2 c f x F_1\left (\frac{3}{2};-n,2;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )\right )}\right )}{15 f^3 x^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((b*x)^(5/2)*(c + d*x)^n)/(e + f*x),x]

[Out]

(2*(b*x)^(5/2)*(c + d*x)^n*((-45*c*e^4*AppellF1[1/2, -n, 1, 3/2, -((d*x)/c), -((
f*x)/e)])/((e + f*x)*(3*c*e*AppellF1[1/2, -n, 1, 3/2, -((d*x)/c), -((f*x)/e)] +
2*d*e*n*x*AppellF1[3/2, 1 - n, 1, 5/2, -((d*x)/c), -((f*x)/e)] - 2*c*f*x*AppellF
1[3/2, -n, 2, 5/2, -((d*x)/c), -((f*x)/e)])) + (15*e^2*Hypergeometric2F1[1/2, -n
, 3/2, -((d*x)/c)] + f*x*(-5*e*Hypergeometric2F1[3/2, -n, 5/2, -((d*x)/c)] + 3*f
*x*Hypergeometric2F1[5/2, -n, 7/2, -((d*x)/c)]))/(1 + (d*x)/c)^n))/(15*f^3*x^2)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) ^{n}}{fx+e} \left ( bx \right ) ^{{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x)^(5/2)*(d*x+c)^n/(f*x+e),x)

[Out]

int((b*x)^(5/2)*(d*x+c)^n/(f*x+e),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(5/2)*(d*x + c)^n/(f*x + e),x, algorithm="maxima")

[Out]

integrate((b*x)^(5/2)*(d*x + c)^n/(f*x + e), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x}{\left (d x + c\right )}^{n} b^{2} x^{2}}{f x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(5/2)*(d*x + c)^n/(f*x + e),x, algorithm="fricas")

[Out]

integral(sqrt(b*x)*(d*x + c)^n*b^2*x^2/(f*x + e), x)

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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+c)**n/(f*x+e),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(5/2)*(d*x + c)^n/(f*x + e),x, algorithm="giac")

[Out]

integrate((b*x)^(5/2)*(d*x + c)^n/(f*x + e), x)