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

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

[Out]

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

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Rubi [A]  time = 0.23358, 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 \[ \frac{2 (b x)^{7/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (\frac{7}{2};-n,2;\frac{9}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{7 b e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.444, size = 48, normalized size = 0.79 \[ \frac{2 \left (b x\right )^{\frac{7}{2}} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (\frac{7}{2},2,- n,\frac{9}{2},- \frac{f x}{e},- \frac{d x}{c} \right )}}{7 b e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.16096, size = 345, normalized size = 5.66 \[ \frac{2 b^2 \sqrt{b x} (c+d x)^n \left (-\frac{9 c e^4 F_1\left (\frac{1}{2};-n,2;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{(e+f x)^2 \left (3 c e F_1\left (\frac{1}{2};-n,2;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+2 d e n x F_1\left (\frac{3}{2};1-n,2;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )-4 c f x F_1\left (\frac{3}{2};-n,3;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )\right )}+\frac{27 c e^3 F_1\left (\frac{1}{2};-n,1;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{(e+f x) \left (3 c e F_1\left (\frac{1}{2};-n,1;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+2 d e n x F_1\left (\frac{3}{2};1-n,1;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )-2 c f x F_1\left (\frac{3}{2};-n,2;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )\right )}+\left (\frac{d x}{c}+1\right )^{-n} \left (f x \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{d x}{c}\right )-6 e \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{d x}{c}\right )\right )\right )}{3 f^3} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*b^2*Sqrt[b*x]*(c + d*x)^n*((27*c*e^3*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*Appell
F1[3/2, -n, 2, 5/2, -((d*x)/c), -((f*x)/e)])) - (9*c*e^4*AppellF1[1/2, -n, 2, 3/
2, -((d*x)/c), -((f*x)/e)])/((e + f*x)^2*(3*c*e*AppellF1[1/2, -n, 2, 3/2, -((d*x
)/c), -((f*x)/e)] + 2*d*e*n*x*AppellF1[3/2, 1 - n, 2, 5/2, -((d*x)/c), -((f*x)/e
)] - 4*c*f*x*AppellF1[3/2, -n, 3, 5/2, -((d*x)/c), -((f*x)/e)])) + (-6*e*Hyperge
ometric2F1[1/2, -n, 3/2, -((d*x)/c)] + f*x*Hypergeometric2F1[3/2, -n, 5/2, -((d*
x)/c)])/(1 + (d*x)/c)^n))/(3*f^3)

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) ^{n}}{ \left ( fx+e \right ) ^{2}} \left ( bx \right ) ^{{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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