3.445 \(\int \frac{(d+e x)^m}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=378 \[ \frac{(d+e x)^{m+1} (b e (1-m)+4 c d)}{2 b^2 d^2 x (b+c x)^2}-\frac{(d+e x)^{m+1} \left (-b^2 e^2 (1-m) m-6 b c d e m+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{2 b^5 d^3 (m+1)}+\frac{c^3 (d+e x)^{m+1} \left (b^2 e^2 \left (m^2-7 m+12\right )-6 b c d e (4-m)+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{2 b^5 (m+1) (c d-b e)^3}+\frac{c (2 c d-b e) (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-6 b c d e+6 c^2 d^2\right )}{2 b^4 d^2 (b+c x) (c d-b e)^2}+\frac{c (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-b c d e (m+4)+6 c^2 d^2\right )}{2 b^3 d^2 (b+c x)^2 (c d-b e)}-\frac{(d+e x)^{m+1}}{2 b d x^2 (b+c x)^2} \]

[Out]

(c*(6*c^2*d^2 - b^2*e^2*(1 - m) - b*c*d*e*(4 + m))*(d + e*x)^(1 + m))/(2*b^3*d^2
*(c*d - b*e)*(b + c*x)^2) - (d + e*x)^(1 + m)/(2*b*d*x^2*(b + c*x)^2) + ((4*c*d
+ b*e*(1 - m))*(d + e*x)^(1 + m))/(2*b^2*d^2*x*(b + c*x)^2) + (c*(2*c*d - b*e)*(
6*c^2*d^2 - 6*b*c*d*e - b^2*e^2*(1 - m))*(d + e*x)^(1 + m))/(2*b^4*d^2*(c*d - b*
e)^2*(b + c*x)) + (c^3*(12*c^2*d^2 - 6*b*c*d*e*(4 - m) + b^2*e^2*(12 - 7*m + m^2
))*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e
)])/(2*b^5*(c*d - b*e)^3*(1 + m)) - ((12*c^2*d^2 - 6*b*c*d*e*m - b^2*e^2*(1 - m)
*m)*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + (e*x)/d])/(2*b^5*d^
3*(1 + m))

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Rubi [A]  time = 1.51904, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{(d+e x)^{m+1} (b e (1-m)+4 c d)}{2 b^2 d^2 x (b+c x)^2}-\frac{(d+e x)^{m+1} \left (-b^2 e^2 (1-m) m-6 b c d e m+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{e x}{d}+1\right )}{2 b^5 d^3 (m+1)}+\frac{c^3 (d+e x)^{m+1} \left (b^2 e^2 \left (m^2-7 m+12\right )-6 b c d e (4-m)+12 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{c (d+e x)}{c d-b e}\right )}{2 b^5 (m+1) (c d-b e)^3}+\frac{c (2 c d-b e) (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-6 b c d e+6 c^2 d^2\right )}{2 b^4 d^2 (b+c x) (c d-b e)^2}+\frac{c (d+e x)^{m+1} \left (-b^2 e^2 (1-m)-b c d e (m+4)+6 c^2 d^2\right )}{2 b^3 d^2 (b+c x)^2 (c d-b e)}-\frac{(d+e x)^{m+1}}{2 b d x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m/(b*x + c*x^2)^3,x]

[Out]

(c*(6*c^2*d^2 - b^2*e^2*(1 - m) - b*c*d*e*(4 + m))*(d + e*x)^(1 + m))/(2*b^3*d^2
*(c*d - b*e)*(b + c*x)^2) - (d + e*x)^(1 + m)/(2*b*d*x^2*(b + c*x)^2) + ((4*c*d
+ b*e*(1 - m))*(d + e*x)^(1 + m))/(2*b^2*d^2*x*(b + c*x)^2) + (c*(2*c*d - b*e)*(
6*c^2*d^2 - 6*b*c*d*e - b^2*e^2*(1 - m))*(d + e*x)^(1 + m))/(2*b^4*d^2*(c*d - b*
e)^2*(b + c*x)) + (c^3*(12*c^2*d^2 - 6*b*c*d*e*(4 - m) + b^2*e^2*(12 - 7*m + m^2
))*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (c*(d + e*x))/(c*d - b*e
)])/(2*b^5*(c*d - b*e)^3*(1 + m)) - ((12*c^2*d^2 - 6*b*c*d*e*m - b^2*e^2*(1 - m)
*m)*(d + e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + (e*x)/d])/(2*b^5*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((e*x+d)**m/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.691492, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (b x+c x^2\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^m/(b*x + c*x^2)^3,x]

[Out]

Integrate[(d + e*x)^m/(b*x + c*x^2)^3, x]

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Maple [F]  time = 0.183, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(c*x^2+b*x)^3,x)

[Out]

int((e*x+d)^m/(c*x^2+b*x)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^m/(c*x^2 + b*x)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3), 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((e*x+d)**m/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*x^2 + b*x)^3,x, algorithm="giac")

[Out]

integrate((e*x + d)^m/(c*x^2 + b*x)^3, x)