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