Optimal. Leaf size=192 \[ -\frac{21 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}-\frac{21 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}+\frac{7 c \sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.135095, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {687, 694, 329, 212, 206, 203} \[ -\frac{21 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}-\frac{21 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt{d} \left (b^2-4 a c\right )^{11/4}}+\frac{7 c \sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\sqrt{b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 687
Rule 694
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx &=-\frac{\sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}-\frac{(7 c) \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{\sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{7 c \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac{\left (21 c^2\right ) \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=-\frac{\sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{7 c \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac{(21 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )}{4 \left (b^2-4 a c\right )^2 d}\\ &=-\frac{\sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{7 c \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac{(21 c) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b^2}{4 c}+\frac{x^4}{4 c d^2}} \, dx,x,\sqrt{d (b+2 c x)}\right )}{2 \left (b^2-4 a c\right )^2 d}\\ &=-\frac{\sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{7 c \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}-\frac{\left (21 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d-x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{\left (21 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d+x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ &=-\frac{\sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac{7 c \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}-\frac{21 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )}{\left (b^2-4 a c\right )^{11/4} \sqrt{d}}-\frac{21 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )}{\left (b^2-4 a c\right )^{11/4} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.379637, size = 179, normalized size = 0.93 \[ \frac{c^2 \left (\frac{\left (b^2-4 a c\right ) (b+2 c x) \left (-c \left (11 a+7 c x^2\right )+b^2-7 b c x\right )}{2 c^2 (a+x (b+c x))^2}+21 \sqrt [4]{b^2-4 a c} \sqrt{b+2 c x} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+21 \sqrt [4]{b^2-4 a c} \sqrt{b+2 c x} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{\left (4 a c-b^2\right )^3 \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.197, size = 419, normalized size = 2.2 \begin{align*} 8\,{\frac{{c}^{2}{d}^{5}\sqrt{2\,cdx+bd}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+14\,{\frac{{c}^{2}{d}^{5}\sqrt{2\,cdx+bd}}{ \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{2} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) }}+{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{4}\ln \left ({ \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{11}{4}}}}+{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{11}{4}}}}-{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{11}{4}}}} \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 = 2.02501, size = 4983, normalized size = 25.95 \begin{align*} \text{result too large to display} \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 = 1.19312, size = 871, normalized size = 4.54 \begin{align*} -\frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} - \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d} - \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} \log \left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} + \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} \log \left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{6} d - 12 \, \sqrt{2} a b^{4} c d + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d - 64 \, \sqrt{2} a^{3} c^{3} d\right )}} - \frac{2 \,{\left (11 \, \sqrt{2 \, c d x + b d} b^{2} c^{2} d^{3} - 44 \, \sqrt{2 \, c d x + b d} a c^{3} d^{3} - 7 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d\right )}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]