Optimal. Leaf size=212 \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 a c C+24 A c^2+7 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.183374, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1661, 640, 612, 621, 206} \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c C+24 A c^2+7 b^2 C\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 a c C+24 A c^2+7 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \left (a+b x+c x^2\right )^{3/2} \left (A+C x^2\right ) \, dx &=\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\int \left (6 A c-a C-\frac{7 b C x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=-\frac{7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\frac{7 b^2 C}{2}+2 c (6 A c-a C)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac{\left (\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\right )\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 )}{512 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2+7 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}-\frac{7 b C \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{C x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (b^2-4 a c\right )^2 \left (24 A c^2+7 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.550195, size = 267, normalized size = 1.26 \[ \frac{\frac{360 A \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{3/2}}+1920 A (b+2 c x) (a+x (b+c x))^{3/2}+\frac{C \left (5 \left (7 b^2-4 a c\right ) \left (\frac{3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )-1792 b (a+x (b+c x))^{5/2}\right )}{c}+2560 C x (a+x (b+c x))^{5/2}}{15360 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 613, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 2.30583, size = 1451, normalized size = 6.84 \begin{align*} \left [\frac{15 \,{\left (7 \, C b^{6} - 60 \, C a b^{4} c + 384 \, A a^{2} c^{4} - 64 \,{\left (C a^{3} + 3 \, A a b^{2}\right )} c^{3} + 24 \,{\left (6 \, C a^{2} b^{2} + A b^{4}\right )} c^{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 ) + 4 \,{\left (1280 \, C c^{6} x^{5} + 1664 \, C b c^{5} x^{4} - 105 \, C b^{5} c + 760 \, C a b^{3} c^{2} + 2400 \, A a b c^{4} - 72 \,{\left (18 \, C a^{2} b + 5 \, A b^{3}\right )} c^{3} + 16 \,{\left (3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}\right )} x^{3} - 8 \,{\left (7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}\right )} x^{2} + 2 \,{\left (35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 2400 \, A a c^{5} + 120 \,{\left (2 \, C a^{2} + A b^{2}\right )} c^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac{15 \,{\left (7 \, C b^{6} - 60 \, C a b^{4} c + 384 \, A a^{2} c^{4} - 64 \,{\left (C a^{3} + 3 \, A a b^{2}\right )} c^{3} + 24 \,{\left (6 \, C a^{2} b^{2} + A b^{4}\right )} c^{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 ) - 2 \,{\left (1280 \, C c^{6} x^{5} + 1664 \, C b c^{5} x^{4} - 105 \, C b^{5} c + 760 \, C a b^{3} c^{2} + 2400 \, A a b c^{4} - 72 \,{\left (18 \, C a^{2} b + 5 \, A b^{3}\right )} c^{3} + 16 \,{\left (3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}\right )} x^{3} - 8 \,{\left (7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}\right )} x^{2} + 2 \,{\left (35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 2400 \, A a c^{5} + 120 \,{\left (2 \, C a^{2} + A b^{2}\right )} c^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + C x^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32111, size = 401, normalized size = 1.89 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, C c x + 13 \, C b\right )} x + \frac{3 \, C b^{2} c^{4} + 140 \, C a c^{5} + 120 \, A c^{6}}{c^{5}}\right )} x - \frac{7 \, C b^{3} c^{3} - 36 \, C a b c^{4} - 360 \, A b c^{5}}{c^{5}}\right )} x + \frac{35 \, C b^{4} c^{2} - 216 \, C a b^{2} c^{3} + 240 \, C a^{2} c^{4} + 120 \, A b^{2} c^{4} + 2400 \, A a c^{5}}{c^{5}}\right )} x - \frac{105 \, C b^{5} c - 760 \, C a b^{3} c^{2} + 1296 \, C a^{2} b c^{3} + 360 \, A b^{3} c^{3} - 2400 \, A a b c^{4}}{c^{5}}\right )} - \frac{{\left (7 \, C b^{6} - 60 \, C a b^{4} c + 144 \, C a^{2} b^{2} c^{2} + 24 \, A b^{4} c^{2} - 64 \, C a^{3} c^{3} - 192 \, A a b^{2} c^{3} + 384 \, A a^{2} c^{4}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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