Optimal. Leaf size=165 \[ -\frac{d^4 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{3/2}}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{48 c}-\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{32 c}+\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0976735, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \[ -\frac{d^4 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{3/2}}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt{a+b x+c x^2}}{48 c}-\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2}}{32 c}+\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 685
Rule 692
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (b d+2 c d x)^4 \sqrt{a+b x+c x^2} \, dx &=\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c}-\frac{\left (b^2-4 a c\right ) \int \frac{(b d+2 c d x)^4}{\sqrt{a+b x+c x^2}} \, dx}{24 c}\\ &=-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{48 c}+\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c}-\frac{\left (\left (b^2-4 a c\right )^2 d^2\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx}{32 c}\\ &=-\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{32 c}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{48 c}+\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c}-\frac{\left (\left (b^2-4 a c\right )^3 d^4\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{64 c}\\ &=-\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{32 c}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{48 c}+\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c}-\frac{\left (\left (b^2-4 a c\right )^3 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 )}{32 c}\\ &=-\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt{a+b x+c x^2}}{32 c}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt{a+b x+c x^2}}{48 c}+\frac{d^4 (b+2 c x)^5 \sqrt{a+b x+c x^2}}{12 c}-\frac{\left (b^2-4 a c\right )^3 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.888537, size = 203, normalized size = 1.23 \[ d^4 \left (\frac{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (2 (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )-\sqrt{c} \sqrt{4 a-\frac{b^2}{c}} \left (4 a c-b^2\right ) \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )\right )}{64 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+\frac{1}{3} (b+2 c x)^3 (a+x (b+c x))^{3/2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.055, size = 413, normalized size = 2.5 \begin{align*}{\frac{7\,{d}^{4}{b}^{3}}{12} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+4\,{d}^{4}{c}^{2}b{x}^{2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}+{\frac{5\,{d}^{4}{b}^{2}cx}{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{4}{b}^{6}}{64}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{d}^{4}cba \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}-2\,{d}^{4}{c}^{2}ax \left ( c{x}^{2}+bx+a \right ) ^{3/2}+{d}^{4}{c}^{2}{a}^{2}x\sqrt{c{x}^{2}+bx+a}+{\frac{c{d}^{4}{a}^{2}b}{2}\sqrt{c{x}^{2}+bx+a}}+{d}^{4}{c}^{{\frac{3}{2}}}{a}^{3}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) +{\frac{8\,{d}^{4}{c}^{3}{x}^{3}}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{4}{b}^{4}x}{16}\sqrt{c{x}^{2}+bx+a}}+{\frac{{d}^{4}{b}^{5}}{32\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{d}^{4}{b}^{3}a}{4}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{d}^{4}{b}^{4}a}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{c{d}^{4}{b}^{2}ax}{2}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{d}^{4}{b}^{2}{a}^{2}}{4}\sqrt{c}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\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 [A] time = 2.27941, size = 1071, normalized size = 6.49 \begin{align*} \left [-\frac{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{c} d^{4} \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 ) - 4 \,{\left (256 \, c^{6} d^{4} x^{5} + 640 \, b c^{5} d^{4} x^{4} + 16 \,{\left (39 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{3} + 8 \,{\left (37 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{2} + 2 \,{\left (31 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} d^{4} x +{\left (3 \, b^{5} c + 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{384 \, c^{2}}, \frac{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-c} d^{4} \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 ) + 2 \,{\left (256 \, c^{6} d^{4} x^{5} + 640 \, b c^{5} d^{4} x^{4} + 16 \,{\left (39 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{3} + 8 \,{\left (37 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{2} + 2 \,{\left (31 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} d^{4} x +{\left (3 \, b^{5} c + 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{4}\right )} \sqrt{c x^{2} + b x + a}}{192 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d^{4} \left (\int b^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 16 c^{4} x^{4} \sqrt{a + b x + c x^{2}}\, dx + \int 32 b c^{3} x^{3} \sqrt{a + b x + c x^{2}}\, dx + \int 24 b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}}\, dx + \int 8 b^{3} c x \sqrt{a + b x + c x^{2}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.28851, size = 350, normalized size = 2.12 \begin{align*} \frac{1}{96} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \, c^{4} d^{4} x + 5 \, b c^{3} d^{4}\right )} x + \frac{39 \, b^{2} c^{7} d^{4} + 4 \, a c^{8} d^{4}}{c^{5}}\right )} x + \frac{37 \, b^{3} c^{6} d^{4} + 12 \, a b c^{7} d^{4}}{c^{5}}\right )} x + \frac{31 \, b^{4} c^{5} d^{4} + 48 \, a b^{2} c^{6} d^{4} - 48 \, a^{2} c^{7} d^{4}}{c^{5}}\right )} x + \frac{3 \, b^{5} c^{4} d^{4} + 32 \, a b^{3} c^{5} d^{4} - 48 \, a^{2} b c^{6} d^{4}}{c^{5}}\right )} + \frac{{\left (b^{6} d^{4} - 12 \, a b^{4} c d^{4} + 48 \, a^{2} b^{2} c^{2} d^{4} - 64 \, a^{3} c^{3} d^{4}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{64 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]