Optimal. Leaf size=151 \[ \frac {e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {e^2 x (3 c d-b e)}{c^2}+\frac {e^3 x^2}{2 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {701, 634, 618, 206, 628} \begin {gather*} \frac {e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {e^2 x (3 c d-b e)}{c^2}+\frac {e^3 x^2}{2 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 628
Rule 634
Rule 701
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx &=\int \left (\frac {e^2 (3 c d-b e)}{c^2}+\frac {e^3 x}{c}+\frac {c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}+\frac {\int \frac {c^2 d^3-3 a c d e^2+a b e^3+e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) x}{a+b x+c x^2} \, dx}{c^2}\\ &=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}+\frac {\left (e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}+\frac {\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^3}\\ &=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}+\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 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^3}\\ &=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {e \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 148, normalized size = 0.98 \begin {gather*} \frac {e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac {2 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^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}}+2 c e^2 x (3 c d-b e)+c^2 e^3 x^2}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^3}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 526, normalized size = 3.48 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{3} x^{2} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} x + {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{3} x^{2} - 2 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} - {\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} e^{3}\right )} x + {\left (3 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 161, normalized size = 1.07 \begin {gather*} \frac {c x^{2} e^{3} + 6 \, c d x e^{2} - 2 \, b x e^{3}}{2 \, c^{2}} + \frac {{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b 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^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 366, normalized size = 2.42 \begin {gather*} \frac {3 a b \,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 {6 a 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 {b^{3} e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {3 b^{2} d \,e^{2} \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^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {e^{3} x^{2}}{2 c}+\frac {2 d^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {a \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}+\frac {b^{2} e^{3} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{3}}-\frac {3 b d \,e^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}-\frac {b \,e^{3} x}{c^{2}}+\frac {3 d^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {3 d \,e^{2} x}{c} \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 = 0.24, size = 208, normalized size = 1.38 \begin {gather*} \frac {e^3\,x^2}{2\,c}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,a^2\,c^2\,e^3-5\,a\,b^2\,c\,e^3+12\,a\,b\,c^2\,d\,e^2-12\,a\,c^3\,d^2\,e+b^4\,e^3-3\,b^3\,c\,d\,e^2+3\,b^2\,c^2\,d^2\,e\right )}{2\,\left (4\,a\,c^4-b^2\,c^3\right )}-x\,\left (\frac {b\,e^3}{c^2}-\frac {3\,d\,e^2}{c}\right )-\frac {\mathrm {atan}\left (\frac {b+2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-b\,c\,d\,e+c^2\,d^2-3\,a\,c\,e^2\right )}{c^3\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 4.31, size = 892, normalized size = 5.91 \begin {gather*} x \left (- \frac {b e^{3}}{c^{2}} + \frac {3 d e^{2}}{c}\right ) + \left (- \frac {e \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {2 a^{2} c e^{3} - a b^{2} e^{3} + 3 a b c d e^{2} + 4 a c^{3} \left (- \frac {e \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )}\right ) - 6 a c^{2} d^{2} e - b^{2} c^{2} \left (- \frac {e \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} - \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )}\right ) + b c^{2} d^{3}}{3 a b c e^{3} - 6 a c^{2} d e^{2} - b^{3} e^{3} + 3 b^{2} c d e^{2} - 3 b c^{2} d^{2} e + 2 c^{3} d^{3}} \right )} + \left (- \frac {e \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {2 a^{2} c e^{3} - a b^{2} e^{3} + 3 a b c d e^{2} + 4 a c^{3} \left (- \frac {e \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )}\right ) - 6 a c^{2} d^{2} e - b^{2} c^{2} \left (- \frac {e \left (a c e^{2} - b^{2} e^{2} + 3 b c d e - 3 c^{2} d^{2}\right )}{2 c^{3}} + \frac {\sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (3 a c e^{2} - b^{2} e^{2} + b c d e - c^{2} d^{2}\right )}{2 c^{3} \left (4 a c - b^{2}\right )}\right ) + b c^{2} d^{3}}{3 a b c e^{3} - 6 a c^{2} d e^{2} - b^{3} e^{3} + 3 b^{2} c d e^{2} - 3 b c^{2} d^{2} e + 2 c^{3} d^{3}} \right )} + \frac {e^{3} x^{2}}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________