Optimal. Leaf size=127 \[ \frac{d x^{m+1} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (m+1)}+\frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,1;1-m;\frac{a}{a+b x}\right )}{b^3 m (a+b x)}+\frac{d^2 x^{m+2} (3 b c-a d)}{b^2 (m+2)}+\frac{d^3 x^{m+3}}{b (m+3)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.250571, antiderivative size = 171, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a b^3 (m+1)}+\frac{d x^{m+1} (b c-a d)^2}{b^3 (m+1)}+\frac{d^2 x^{m+2} (b c-a d)}{b^2 (m+2)}+\frac{c d x^{m+1} (b c-a d)}{b^2 (m+1)}+\frac{c^2 d x^{m+1}}{b (m+1)}+\frac{2 c d^2 x^{m+2}}{b (m+2)}+\frac{d^3 x^{m+3}}{b (m+3)} \]
Antiderivative was successfully verified.
[In] Int[(x^m*(c + d*x)^3)/(a + b*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 39.2944, size = 144, normalized size = 1.13 \[ \frac{c^{2} d x^{m + 1}}{b \left (m + 1\right )} + \frac{2 c d^{2} x^{m + 2}}{b \left (m + 2\right )} + \frac{d^{3} x^{m + 3}}{b \left (m + 3\right )} - \frac{c d x^{m + 1} \left (a d - b c\right )}{b^{2} \left (m + 1\right )} - \frac{d^{2} x^{m + 2} \left (a d - b c\right )}{b^{2} \left (m + 2\right )} + \frac{d x^{m + 1} \left (a d - b c\right )^{2}}{b^{3} \left (m + 1\right )} - \frac{x^{m + 1} \left (a d - b c\right )^{3}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a b^{3} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(d*x+c)**3/(b*x+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.227463, size = 113, normalized size = 0.89 \[ \frac{x^{m+1} \left (\frac{c^3 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{m+1}+d x \left (\frac{3 c^2 \, _2F_1\left (1,m+2;m+3;-\frac{b x}{a}\right )}{m+2}+d x \left (\frac{3 c \, _2F_1\left (1,m+3;m+4;-\frac{b x}{a}\right )}{m+3}+\frac{d x \, _2F_1\left (1,m+4;m+5;-\frac{b x}{a}\right )}{m+4}\right )\right )\right )}{a} \]
Antiderivative was successfully verified.
[In] Integrate[(x^m*(c + d*x)^3)/(a + b*x),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( dx+c \right ) ^{3}}{bx+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(d*x+c)^3/(b*x+a),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{3} x^{m}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*x^m/(b*x + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} x^{m}}{b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*x^m/(b*x + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 10.9081, size = 303, normalized size = 2.39 \[ \frac{c^{3} m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{c^{3} x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{3 c^{2} d m x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{6 c^{2} d x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{3 c d^{2} m x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac{9 c d^{2} x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac{d^{3} m x^{4} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} + \frac{4 d^{3} x^{4} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(d*x+c)**3/(b*x+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{3} x^{m}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*x^m/(b*x + a),x, algorithm="giac")
[Out]