3.943 \(\int \frac{(b x)^m (c+d x)^n}{(e+f x)^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.102971, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e^2 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.6826, size = 48, normalized size = 0.76 \[ \frac{\left (b x\right )^{m + 1} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,2,- n,m + 2,- \frac{f x}{e},- \frac{d x}{c} \right )}}{b e^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.415252, size = 153, normalized size = 2.43 \[ \frac{c e (m+2) x (b x)^m (c+d x)^n F_1\left (m+1;-n,2;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{(m+1) (e+f x)^2 \left (c e (m+2) F_1\left (m+1;-n,2;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )+x \left (d e n F_1\left (m+2;1-n,2;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )-2 c f F_1\left (m+2;-n,3;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(c*e*(2 + m)*x*(b*x)^m*(c + d*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((d*x)/c), -((
f*x)/e)])/((1 + m)*(e + f*x)^2*(c*e*(2 + m)*AppellF1[1 + m, -n, 2, 2 + m, -((d*x
)/c), -((f*x)/e)] + x*(d*e*n*AppellF1[2 + m, 1 - n, 2, 3 + m, -((d*x)/c), -((f*x
)/e)] - 2*c*f*AppellF1[2 + m, -n, 3, 3 + m, -((d*x)/c), -((f*x)/e)])))

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n}}{ \left ( fx+e \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x)^m*(d*x + c)^n/(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)**m*(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. \[ \int \frac{\left (b x\right )^{m}{\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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