Optimal. Leaf size=123 \[ 96 c^2 d^9 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+96 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^9 (b+2 c x)^4 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0848583, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {686, 692, 628} \[ 96 c^2 d^9 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+96 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^9 (b+2 c x)^4 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 686
Rule 692
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+\left (8 c d^2\right ) \int \frac{(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 d^4\right ) \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=48 c^2 d^9 (b+2 c x)^4-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac{d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac{8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+96 c^2 \left (b^2-4 a c\right )^2 d^9 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0610094, size = 131, normalized size = 1.07 \[ d^9 \left (-384 c^4 x^2 \left (2 a c-b^2\right )+256 b c^3 x \left (b^2-3 a c\right )+96 c^2 \left (b^2-4 a c\right )^2 \log (a+x (b+c x))+\frac{16 c \left (4 a c-b^2\right )^3}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^4}{2 (a+x (b+c x))^2}+256 b c^5 x^3+128 c^6 x^4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.052, size = 465, normalized size = 3.8 \begin{align*} 128\,{d}^{9}{c}^{6}{x}^{4}+256\,{d}^{9}b{c}^{5}{x}^{3}-768\,{d}^{9}{x}^{2}a{c}^{5}+384\,{d}^{9}{x}^{2}{b}^{2}{c}^{4}-768\,{d}^{9}ab{c}^{4}x+256\,{d}^{9}{b}^{3}{c}^{3}x+1024\,{\frac{{d}^{9}{x}^{2}{a}^{3}{c}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-768\,{\frac{{d}^{9}{x}^{2}{a}^{2}{b}^{2}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+192\,{\frac{{d}^{9}{x}^{2}a{b}^{4}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-16\,{\frac{{d}^{9}{x}^{2}{b}^{6}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+1024\,{\frac{{d}^{9}x{a}^{3}b{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-768\,{\frac{{d}^{9}x{a}^{2}{b}^{3}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+192\,{\frac{{d}^{9}xa{b}^{5}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-16\,{\frac{{d}^{9}x{b}^{7}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+896\,{\frac{{d}^{9}{a}^{4}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-640\,{\frac{{d}^{9}{a}^{3}{b}^{2}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+144\,{\frac{{d}^{9}{a}^{2}{b}^{4}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{{d}^{9}a{b}^{6}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{9}{b}^{8}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+1536\,{d}^{9}\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{c}^{4}-768\,{d}^{9}\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}{c}^{3}+96\,{d}^{9}\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}{c}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.17008, size = 375, normalized size = 3.05 \begin{align*} 128 \, c^{6} d^{9} x^{4} + 256 \, b c^{5} d^{9} x^{3} + 384 \,{\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{9} x^{2} + 256 \,{\left (b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{9} x + 96 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{9} \log \left (c x^{2} + b x + a\right ) - \frac{32 \,{\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{9} x^{2} + 32 \,{\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{9} x +{\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.98246, size = 1035, normalized size = 8.41 \begin{align*} \frac{256 \, c^{8} d^{9} x^{8} + 1024 \, b c^{7} d^{9} x^{7} + 1024 \,{\left (2 \, b^{2} c^{6} - a c^{7}\right )} d^{9} x^{6} + 512 \,{\left (5 \, b^{3} c^{5} - 6 \, a b c^{6}\right )} d^{9} x^{5} + 256 \,{\left (7 \, b^{4} c^{4} - 8 \, a b^{2} c^{5} - 11 \, a^{2} c^{6}\right )} d^{9} x^{4} + 512 \,{\left (b^{5} c^{3} + 2 \, a b^{3} c^{4} - 11 \, a^{2} b c^{5}\right )} d^{9} x^{3} - 32 \,{\left (b^{6} c^{2} - 44 \, a b^{4} c^{3} + 120 \, a^{2} b^{2} c^{4} - 16 \, a^{3} c^{5}\right )} d^{9} x^{2} - 32 \,{\left (b^{7} c - 12 \, a b^{5} c^{2} + 32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b c^{4}\right )} d^{9} x -{\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9} + 192 \,{\left ({\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{9} x^{4} + 2 \,{\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{9} x^{3} +{\left (b^{6} c^{2} - 6 \, a b^{4} c^{3} + 32 \, a^{3} c^{5}\right )} d^{9} x^{2} + 2 \,{\left (a b^{5} c^{2} - 8 \, a^{2} b^{3} c^{3} + 16 \, a^{3} b c^{4}\right )} d^{9} x +{\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} d^{9}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 31.3321, size = 320, normalized size = 2.6 \begin{align*} 256 b c^{5} d^{9} x^{3} + 128 c^{6} d^{9} x^{4} + 96 c^{2} d^{9} \left (4 a c - b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} + x^{2} \left (- 768 a c^{5} d^{9} + 384 b^{2} c^{4} d^{9}\right ) + x \left (- 768 a b c^{4} d^{9} + 256 b^{3} c^{3} d^{9}\right ) + \frac{1792 a^{4} c^{4} d^{9} - 1280 a^{3} b^{2} c^{3} d^{9} + 288 a^{2} b^{4} c^{2} d^{9} - 16 a b^{6} c d^{9} - b^{8} d^{9} + x^{2} \left (2048 a^{3} c^{5} d^{9} - 1536 a^{2} b^{2} c^{4} d^{9} + 384 a b^{4} c^{3} d^{9} - 32 b^{6} c^{2} d^{9}\right ) + x \left (2048 a^{3} b c^{4} d^{9} - 1536 a^{2} b^{3} c^{3} d^{9} + 384 a b^{5} c^{2} d^{9} - 32 b^{7} c d^{9}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.27181, size = 404, normalized size = 3.28 \begin{align*} 96 \,{\left (b^{4} c^{2} d^{9} - 8 \, a b^{2} c^{3} d^{9} + 16 \, a^{2} c^{4} d^{9}\right )} \log \left (c x^{2} + b x + a\right ) - \frac{b^{8} d^{9} + 16 \, a b^{6} c d^{9} - 288 \, a^{2} b^{4} c^{2} d^{9} + 1280 \, a^{3} b^{2} c^{3} d^{9} - 1792 \, a^{4} c^{4} d^{9} + 32 \,{\left (b^{6} c^{2} d^{9} - 12 \, a b^{4} c^{3} d^{9} + 48 \, a^{2} b^{2} c^{4} d^{9} - 64 \, a^{3} c^{5} d^{9}\right )} x^{2} + 32 \,{\left (b^{7} c d^{9} - 12 \, a b^{5} c^{2} d^{9} + 48 \, a^{2} b^{3} c^{3} d^{9} - 64 \, a^{3} b c^{4} d^{9}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (c^{18} d^{9} x^{4} + 2 \, b c^{17} d^{9} x^{3} + 3 \, b^{2} c^{16} d^{9} x^{2} - 6 \, a c^{17} d^{9} x^{2} + 2 \, b^{3} c^{15} d^{9} x - 6 \, a b c^{16} d^{9} x\right )}}{c^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]