Optimal. Leaf size=126 \[ -\frac {3 e \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {3 e^2 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}-\frac {(d+e x)^3}{a+b x+c x^2}+\frac {3 e^3 x}{c} \]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {768, 701, 634, 618, 206, 628} \begin {gather*} -\frac {3 e \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {3 e^2 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}-\frac {(d+e x)^3}{a+b x+c x^2}+\frac {3 e^3 x}{c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 628
Rule 634
Rule 701
Rule 768
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^3}{a+b x+c x^2}+(3 e) \int \frac {(d+e x)^2}{a+b x+c x^2} \, dx\\ &=-\frac {(d+e x)^3}{a+b x+c x^2}+(3 e) \int \left (\frac {e^2}{c}+\frac {c d^2-a e^2+e (2 c d-b e) x}{c \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {3 e^3 x}{c}-\frac {(d+e x)^3}{a+b x+c x^2}+\frac {(3 e) \int \frac {c d^2-a e^2+e (2 c d-b e) x}{a+b x+c x^2} \, dx}{c}\\ &=\frac {3 e^3 x}{c}-\frac {(d+e x)^3}{a+b x+c x^2}+\frac {\left (3 e^2 (2 c d-b e)\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}+\frac {\left (3 e \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^2}\\ &=\frac {3 e^3 x}{c}-\frac {(d+e x)^3}{a+b x+c x^2}+\frac {3 e^2 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}-\frac {\left (3 e \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2}\\ &=\frac {3 e^3 x}{c}-\frac {(d+e x)^3}{a+b x+c x^2}-\frac {3 e \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {3 e^2 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 161, normalized size = 1.28 \begin {gather*} \frac {\frac {6 e \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-\frac {2 \left (-c e^2 (3 a d+a e x+3 b d x)+b e^3 (a+b x)+c^2 d^2 (d+3 e x)\right )}{a+x (b+c x)}-3 e^2 (b e-2 c d) \log (a+x (b+c x))+4 c e^3 x}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 1050, normalized size = 8.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 173, normalized size = 1.37 \begin {gather*} \frac {2 \, x e^{3}}{c} + \frac {3 \, {\left (2 \, c d e^{2} - b e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {3 \, {\left (2 \, c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3} - 2 \, a c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{2}} - \frac {c^{2} d^{3} - 3 \, a c d e^{2} + a b e^{3} + {\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )} x}{{\left (c x^{2} + b x + a\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 363, normalized size = 2.88 \begin {gather*} \frac {a \,e^{3} x}{\left (c \,x^{2}+b x +a \right ) c}-\frac {6 a \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {b^{2} e^{3} x}{\left (c \,x^{2}+b x +a \right ) c^{2}}+\frac {3 b^{2} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {3 b d \,e^{2} x}{\left (c \,x^{2}+b x +a \right ) c}-\frac {6 b d \,e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {3 d^{2} e x}{c \,x^{2}+b x +a}+\frac {6 d^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {a b \,e^{3}}{\left (c \,x^{2}+b x +a \right ) c^{2}}+\frac {3 a d \,e^{2}}{\left (c \,x^{2}+b x +a \right ) c}-\frac {3 b \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}+\frac {3 d \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {2 e^{3} x}{c}-\frac {d^{3}}{c \,x^{2}+b x +a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.95, size = 226, normalized size = 1.79 \begin {gather*} \frac {2\,e^3\,x}{c}-\frac {\frac {c^2\,d^3-3\,a\,c\,d\,e^2+a\,b\,e^3}{c}+\frac {x\,\left (b^2\,e^3-3\,b\,c\,d\,e^2+3\,c^2\,d^2\,e-a\,c\,e^3\right )}{c}}{c^2\,x^2+b\,c\,x+a\,c}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (3\,b^3\,e^3-6\,d\,b^2\,c\,e^2-12\,a\,b\,c\,e^3+24\,a\,d\,c^2\,e^2\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )}+\frac {3\,e\,\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2-2\,a\,c\,e^2\right )}{c^2\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 11.14, size = 733, normalized size = 5.82 \begin {gather*} \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} - \frac {3 e \sqrt {- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 3 a b e^{3} - 4 a c^{2} \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} - \frac {3 e \sqrt {- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + 12 a c d e^{2} + b^{2} c \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} - \frac {3 e \sqrt {- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) - 3 b c d^{2} e}{6 a c e^{3} - 3 b^{2} e^{3} + 6 b c d e^{2} - 6 c^{2} d^{2} e} \right )} + \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} + \frac {3 e \sqrt {- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- 3 a b e^{3} - 4 a c^{2} \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} + \frac {3 e \sqrt {- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + 12 a c d e^{2} + b^{2} c \left (- \frac {3 e^{2} \left (b e - 2 c d\right )}{2 c^{2}} + \frac {3 e \sqrt {- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) - 3 b c d^{2} e}{6 a c e^{3} - 3 b^{2} e^{3} + 6 b c d e^{2} - 6 c^{2} d^{2} e} \right )} + \frac {- a b e^{3} + 3 a c d e^{2} - c^{2} d^{3} + x \left (a c e^{3} - b^{2} e^{3} + 3 b c d e^{2} - 3 c^{2} d^{2} e\right )}{a c^{2} + b c^{2} x + c^{3} x^{2}} + \frac {2 e^{3} x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________