Optimal. Leaf size=100 \[ 20 c d^6 \left (b^2-4 a c\right ) (b+2 c x)-20 c d^6 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}+\frac {20}{3} c d^6 (b+2 c x)^3 \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \begin {gather*} 20 c d^6 \left (b^2-4 a c\right ) (b+2 c x)-20 c d^6 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}+\frac {20}{3} c d^6 (b+2 c x)^3 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 686
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}+\left (10 c d^2\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac {20}{3} c d^6 (b+2 c x)^3-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}+\left (10 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=20 c \left (b^2-4 a c\right ) d^6 (b+2 c x)+\frac {20}{3} c d^6 (b+2 c x)^3-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}+\left (10 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=20 c \left (b^2-4 a c\right ) d^6 (b+2 c x)+\frac {20}{3} c d^6 (b+2 c x)^3-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}-\left (20 c \left (b^2-4 a c\right )^2 d^6\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=20 c \left (b^2-4 a c\right ) d^6 (b+2 c x)+\frac {20}{3} c d^6 (b+2 c x)^3-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}-20 c \left (b^2-4 a c\right )^{3/2} d^6 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 108, normalized size = 1.08 \begin {gather*} d^6 \left (-16 c^2 x \left (8 a c-3 b^2\right )-\frac {\left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}+20 c \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+32 b c^3 x^2+\frac {64 c^4 x^3}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 504, normalized size = 5.04 \begin {gather*} \left [\frac {64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \, {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \, {\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 30 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\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 )}{3 \, {\left (c x^{2} + b x + a\right )}}, \frac {64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \, {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \, {\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 60 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\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 )}{3 \, {\left (c x^{2} + b x + a\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 197, normalized size = 1.97 \begin {gather*} \frac {20 \, {\left (b^{4} c d^{6} - 8 \, a b^{2} c^{2} d^{6} + 16 \, a^{2} c^{3} d^{6}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{4} c d^{6} x - 16 \, a b^{2} c^{2} d^{6} x + 32 \, a^{2} c^{3} d^{6} x + b^{5} d^{6} - 8 \, a b^{3} c d^{6} + 16 \, a^{2} b c^{2} d^{6}}{c x^{2} + b x + a} + \frac {16 \, {\left (4 \, c^{10} d^{6} x^{3} + 6 \, b c^{9} d^{6} x^{2} + 9 \, b^{2} c^{8} d^{6} x - 24 \, a c^{9} d^{6} x\right )}}{3 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 312, normalized size = 3.12 \begin {gather*} \frac {64 c^{4} d^{6} x^{3}}{3}-\frac {32 a^{2} c^{3} d^{6} x}{c \,x^{2}+b x +a}+\frac {320 a^{2} c^{3} d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {16 a \,b^{2} c^{2} d^{6} x}{c \,x^{2}+b x +a}-\frac {160 a \,b^{2} c^{2} d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {2 b^{4} c \,d^{6} x}{c \,x^{2}+b x +a}+\frac {20 b^{4} c \,d^{6} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+32 b \,c^{3} d^{6} x^{2}-\frac {16 a^{2} b \,c^{2} d^{6}}{c \,x^{2}+b x +a}+\frac {8 a \,b^{3} c \,d^{6}}{c \,x^{2}+b x +a}-128 a \,c^{3} d^{6} x -\frac {b^{5} d^{6}}{c \,x^{2}+b x +a}+48 b^{2} c^{2} d^{6} x \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 = 0.08, size = 230, normalized size = 2.30 \begin {gather*} \frac {64\,c^4\,d^6\,x^3}{3}-\frac {x\,\left (32\,a^2\,c^3\,d^6-16\,a\,b^2\,c^2\,d^6+2\,b^4\,c\,d^6\right )+b^5\,d^6+16\,a^2\,b\,c^2\,d^6-8\,a\,b^3\,c\,d^6}{c\,x^2+b\,x+a}-x\,\left (64\,c^2\,d^6\,\left (b^2+2\,a\,c\right )-112\,b^2\,c^2\,d^6\right )+32\,b\,c^3\,d^6\,x^2+20\,c\,d^6\,\mathrm {atan}\left (\frac {20\,c^2\,d^6\,x\,{\left (4\,a\,c-b^2\right )}^{3/2}+10\,b\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^{3/2}}{160\,a^2\,c^3\,d^6-80\,a\,b^2\,c^2\,d^6+10\,b^4\,c\,d^6}\right )\,{\left (4\,a\,c-b^2\right )}^{3/2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.64, size = 313, normalized size = 3.13 \begin {gather*} 32 b c^{3} d^{6} x^{2} + \frac {64 c^{4} d^{6} x^{3}}{3} - 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {40 a b c^{2} d^{6} - 10 b^{3} c d^{6} - 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {40 a b c^{2} d^{6} - 10 b^{3} c d^{6} + 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + x \left (- 128 a c^{3} d^{6} + 48 b^{2} c^{2} d^{6}\right ) + \frac {- 16 a^{2} b c^{2} d^{6} + 8 a b^{3} c d^{6} - b^{5} d^{6} + x \left (- 32 a^{2} c^{3} d^{6} + 16 a b^{2} c^{2} d^{6} - 2 b^{4} c d^{6}\right )}{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________