Optimal. Leaf size=96 \[ 16 c^{3/2} d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0499499, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {686, 621, 206} \[ 16 c^{3/2} d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 686
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}+\left (4 c d^2\right ) \int \frac{(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}+\left (16 c^2 d^4\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}+\left (32 c^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )\\ &=-\frac{2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{8 c d^4 (b+2 c x)}{\sqrt{a+b x+c x^2}}+16 c^{3/2} d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.348217, size = 142, normalized size = 1.48 \[ d^4 \left (\frac{16 c^{3/2} \sqrt{a+x (b+c x)} \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )}{\sqrt{4 a-\frac{b^2}{c}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}-\frac{2 (b+2 c x) \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )}{3 (a+x (b+c x))^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.053, size = 531, normalized size = 5.5 \begin{align*} -8\,{\frac{{c}^{2}{d}^{4}{b}^{2}ax}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}-64\,{\frac{{c}^{3}{d}^{4}{b}^{2}ax}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}-24\,{\frac{{d}^{4}b{c}^{2}{x}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}-18\,{\frac{{d}^{4}{b}^{2}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}+8\,{\frac{{b}^{5}{d}^{4}c}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}-16\,{\frac{c{d}^{4}ba}{ \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}+8\,{\frac{c{d}^{4}{b}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{{d}^{4}{b}^{3}}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{c{d}^{4}{b}^{4}x}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}+16\,{\frac{{c}^{2}{d}^{4}{b}^{4}x}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{c{d}^{4}{b}^{3}a}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3/2}}}-32\,{\frac{{c}^{2}{d}^{4}{b}^{3}a}{ \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{c{x}^{2}+bx+a}}}+16\,{\frac{{c}^{2}{d}^{4}{b}^{2}x}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{16\,{d}^{4}{c}^{3}{x}^{3}}{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}+{\frac{{b}^{5}{d}^{4}}{4\,ac-{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}}-16\,{\frac{{d}^{4}{c}^{2}x}{\sqrt{c{x}^{2}+bx+a}}}+8\,{\frac{c{d}^{4}b}{\sqrt{c{x}^{2}+bx+a}}}+16\,{d}^{4}{c}^{3/2}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 8.80002, size = 979, normalized size = 10.2 \begin{align*} \left [\frac{2 \,{\left (12 \,{\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} +{\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) -{\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x +{\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\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 )}}, -\frac{2 \,{\left (24 \,{\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} +{\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) +{\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \,{\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x +{\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\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 )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17013, size = 458, normalized size = 4.77 \begin{align*} -\frac{8 \, d^{4} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} - \frac{2 \,{\left (8 \,{\left (\frac{2 \,{\left (b^{4} c^{3} d^{4} - 8 \, a b^{2} c^{4} d^{4} + 16 \, a^{2} c^{5} d^{4}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (b^{5} c^{2} d^{4} - 8 \, a b^{3} c^{3} d^{4} + 16 \, a^{2} b c^{4} d^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (3 \, b^{6} c d^{4} - 20 \, a b^{4} c^{2} d^{4} + 16 \, a^{2} b^{2} c^{3} d^{4} + 64 \, a^{3} c^{4} d^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{7} d^{4} + 4 \, a b^{5} c d^{4} - 80 \, a^{2} b^{3} c^{2} d^{4} + 192 \, a^{3} b c^{3} d^{4}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]