3.1851 \(\int \frac{(a+b x)^m}{(c+d x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0390526, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m/(c + d*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.98509, size = 42, normalized size = 0.78 \[ - \frac{b^{2} \left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{\left (m + 1\right ) \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2*(a + b*x)**(m + 1)*hyper((3, m + 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/((
m + 1)*(a*d - b*c)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.054512, size = 0, normalized size = 0. \[ \int \frac{(a+b x)^m}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(a + b*x)^m/(c + d*x)^3, x]

_______________________________________________________________________________________

Maple [F]  time = 0.107, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/(d*x+c)^3,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m/(d*x + c)^3, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^m/(d*x + c)^3, x)