3.354 \(\int \frac{x^m}{(a+b x) (c+d x)^3} \, dx\)

Optimal. Leaf size=206 \[ \frac{d x^{m+1} \left (a^2 d^2 (1-m) m-2 a b c d (2-m) m-b^2 c^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )}{2 c^3 (m+1) (b c-a d)^3}+\frac{b^3 x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) (b c-a d)^3}+\frac{d x^{m+1} (a d (1-m)-b c (3-m))}{2 c^2 (c+d x) (b c-a d)^2}-\frac{d x^{m+1}}{2 c (c+d x)^2 (b c-a d)} \]

[Out]

-(d*x^(1 + m))/(2*c*(b*c - a*d)*(c + d*x)^2) + (d*(a*d*(1 - m) - b*c*(3 - m))*x^
(1 + m))/(2*c^2*(b*c - a*d)^2*(c + d*x)) + (b^3*x^(1 + m)*Hypergeometric2F1[1, 1
+ m, 2 + m, -((b*x)/a)])/(a*(b*c - a*d)^3*(1 + m)) + (d*(a^2*d^2*(1 - m)*m - 2*
a*b*c*d*(2 - m)*m - b^2*c^2*(2 - 3*m + m^2))*x^(1 + m)*Hypergeometric2F1[1, 1 +
m, 2 + m, -((d*x)/c)])/(2*c^3*(b*c - a*d)^3*(1 + m))

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Rubi [A]  time = 0.685384, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{d x^{m+1} \left (a^2 d^2 (1-m) m-2 a b c d (2-m) m-b^2 c^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (1,m+1;m+2;-\frac{d x}{c}\right )}{2 c^3 (m+1) (b c-a d)^3}+\frac{b^3 x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a (m+1) (b c-a d)^3}+\frac{d x^{m+1} (a d (1-m)-b c (3-m))}{2 c^2 (c+d x) (b c-a d)^2}-\frac{d x^{m+1}}{2 c (c+d x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/((a + b*x)*(c + d*x)^3),x]

[Out]

-(d*x^(1 + m))/(2*c*(b*c - a*d)*(c + d*x)^2) + (d*(a*d*(1 - m) - b*c*(3 - m))*x^
(1 + m))/(2*c^2*(b*c - a*d)^2*(c + d*x)) + (b^3*x^(1 + m)*Hypergeometric2F1[1, 1
+ m, 2 + m, -((b*x)/a)])/(a*(b*c - a*d)^3*(1 + m)) + (d*(a^2*d^2*(1 - m)*m - 2*
a*b*c*d*(2 - m)*m - b^2*c^2*(2 - 3*m + m^2))*x^(1 + m)*Hypergeometric2F1[1, 1 +
m, 2 + m, -((d*x)/c)])/(2*c^3*(b*c - a*d)^3*(1 + m))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(b*x+a)/(d*x+c)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.375888, size = 142, normalized size = 0.69 \[ \frac{a c (m+2) x^{m+1} F_1\left (m+1;3,1;m+2;-\frac{d x}{c},-\frac{b x}{a}\right )}{(m+1) (a+b x) (c+d x)^3 \left (a c (m+2) F_1\left (m+1;3,1;m+2;-\frac{d x}{c},-\frac{b x}{a}\right )-x \left (b c F_1\left (m+2;3,2;m+3;-\frac{d x}{c},-\frac{b x}{a}\right )+3 a d F_1\left (m+2;4,1;m+3;-\frac{d x}{c},-\frac{b x}{a}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^m/((a + b*x)*(c + d*x)^3),x]

[Out]

(a*c*(2 + m)*x^(1 + m)*AppellF1[1 + m, 3, 1, 2 + m, -((d*x)/c), -((b*x)/a)])/((1
+ m)*(a + b*x)*(c + d*x)^3*(a*c*(2 + m)*AppellF1[1 + m, 3, 1, 2 + m, -((d*x)/c)
, -((b*x)/a)] - x*(b*c*AppellF1[2 + m, 3, 2, 3 + m, -((d*x)/c), -((b*x)/a)] + 3*
a*d*AppellF1[2 + m, 4, 1, 3 + m, -((d*x)/c), -((b*x)/a)])))

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( bx+a \right ) \left ( dx+c \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x+a)/(d*x+c)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/((b*x + a)*(d*x + c)^3),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b d^{3} x^{4} + a c^{3} +{\left (3 \, b c d^{2} + a d^{3}\right )} x^{3} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} x^{2} +{\left (b c^{3} + 3 \, a c^{2} d\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/((b*x + a)*(d*x + c)^3),x, algorithm="fricas")

[Out]

integral(x^m/(b*d^3*x^4 + a*c^3 + (3*b*c*d^2 + a*d^3)*x^3 + 3*(b*c^2*d + a*c*d^2
)*x^2 + (b*c^3 + 3*a*c^2*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x + a\right )}{\left (d x + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/((b*x + a)*(d*x + c)^3),x, algorithm="giac")

[Out]

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