3.353 \(\int \frac{x^m}{(a+b x) (c+d x)^2} \, dx\)

Optimal. Leaf size=125 \[ \frac{b^2 x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) (b c-a d)^2}-\frac{d x^{m+1} (a d m+b c (1-m)) \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )}{c^2 (m+1) (b c-a d)^2}-\frac{d x^{m+1}}{c (c+d x) (b c-a d)} \]

[Out]

-((d*x^(1 + m))/(c*(b*c - a*d)*(c + d*x))) + (b^2*x^(1 + m)*Hypergeometric2F1[1,
1 + m, 2 + m, -((b*x)/a)])/(a*(b*c - a*d)^2*(1 + m)) - (d*(b*c*(1 - m) + a*d*m)
*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*x)/c)])/(c^2*(b*c - a*d)^2*(1
+ m))

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Rubi [A]  time = 0.291238, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{b^2 x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) (b c-a d)^2}-\frac{d x^{m+1} (a d m+b (c-c m)) \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )}{c^2 (m+1) (b c-a d)^2}-\frac{d x^{m+1}}{c (c+d x) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-((d*x^(1 + m))/(c*(b*c - a*d)*(c + d*x))) + (b^2*x^(1 + m)*Hypergeometric2F1[1,
1 + m, 2 + m, -((b*x)/a)])/(a*(b*c - a*d)^2*(1 + m)) - (d*(a*d*m + b*(c - c*m))
*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*x)/c)])/(c^2*(b*c - a*d)^2*(1
+ m))

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Rubi in Sympy [A]  time = 50.2244, size = 100, normalized size = 0.8 \[ \frac{d x^{m + 1}}{c \left (c + d x\right ) \left (a d - b c\right )} - \frac{d x^{m + 1} \left (a d m - b c m + b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{c^{2} \left (m + 1\right ) \left (a d - b c\right )^{2}} + \frac{b^{2} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a \left (m + 1\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x**(m + 1)/(c*(c + d*x)*(a*d - b*c)) - d*x**(m + 1)*(a*d*m - b*c*m + b*c)*hype
r((1, m + 1), (m + 2,), -d*x/c)/(c**2*(m + 1)*(a*d - b*c)**2) + b**2*x**(m + 1)*
hyper((1, m + 1), (m + 2,), -b*x/a)/(a*(m + 1)*(a*d - b*c)**2)

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Mathematica [C]  time = 0.349384, size = 142, normalized size = 1.14 \[ \frac{a c (m+2) x^{m+1} F_1\left (m+1;2,1;m+2;-\frac{d x}{c},-\frac{b x}{a}\right )}{(m+1) (a+b x) (c+d x)^2 \left (a c (m+2) F_1\left (m+1;2,1;m+2;-\frac{d x}{c},-\frac{b x}{a}\right )-x \left (b c F_1\left (m+2;2,2;m+3;-\frac{d x}{c},-\frac{b x}{a}\right )+2 a d F_1\left (m+2;3,1;m+3;-\frac{d x}{c},-\frac{b x}{a}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( bx+a \right ) \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x+a)/(d*x+c)^2,x)

[Out]

int(x^m/(b*x+a)/(d*x+c)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/((b*x + a)*(d*x + c)^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b d^{2} x^{3} + a c^{2} +{\left (2 \, b c d + a d^{2}\right )} x^{2} +{\left (b c^{2} + 2 \, a c d\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b*d^2*x^3 + a*c^2 + (2*b*c*d + a*d^2)*x^2 + (b*c^2 + 2*a*c*d)*x),
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(x**m/(b*x+a)/(d*x+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/((b*x + a)*(d*x + c)^2), x)