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

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

[Out]

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

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Rubi [A]  time = 0.102751, 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 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.4671, size = 46, normalized size = 0.73 \[ \frac{\left (b x\right )^{m + 1} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{f x}{e},- \frac{d x}{c} \right )}}{b e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.402652, size = 153, normalized size = 2.43 \[ \frac{c e (m+2) x (b x)^m (c+d x)^n F_1\left (m+1;-n,1;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{(m+1) (e+f x) \left (c e (m+2) F_1\left (m+1;-n,1;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )+x \left (d e n F_1\left (m+2;1-n,1;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )-c f F_1\left (m+2;-n,2;m+3;-\frac{d x}{c},-\frac{f x}{e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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