Optimal. Leaf size=252 \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{384 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{6144 c^4}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a B c-16 A b c+9 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}-\frac{\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]
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Rubi [A] time = 0.124824, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {779, 612, 621, 206} \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{384 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{6144 c^4}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a B c-16 A b c+9 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}-\frac{\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]
Antiderivative was successfully verified.
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Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=-\frac{(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{\left (9 b^2 B-16 A b c-4 a B c\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{32 c^2}\\ &=\frac{\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac{\left (5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{\left (5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{4096 c^4}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac{\left (5 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B c\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac{\left (5 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B 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 )}{16384 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac{5 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.395813, size = 206, normalized size = 0.82 \[ \frac{\frac{7 \left (-2 a B c-8 A b c+\frac{9 b^2 B}{2}\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{3072 c^{7/2}}+(a+x (b+c x))^{7/2} (2 c (8 A+7 B x)-9 b B)}{112 c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 1034, normalized size = 4.1 \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 [B] time = 2.32495, size = 2493, normalized size = 9.89 \begin{align*} \text{result too large to display} \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 x \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21061, size = 713, normalized size = 2.83 \begin{align*} \frac{1}{344064} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, B c^{2} x + \frac{33 \, B b c^{8} + 16 \, A c^{9}}{c^{7}}\right )} x + \frac{243 \, B b^{2} c^{7} + 476 \, B a c^{8} + 464 \, A b c^{8}}{c^{7}}\right )} x + \frac{3 \, B b^{3} c^{6} + 1228 \, B a b c^{7} + 592 \, A b^{2} c^{7} + 1152 \, A a c^{8}}{c^{7}}\right )} x - \frac{27 \, B b^{4} c^{5} - 216 \, B a b^{2} c^{6} - 48 \, A b^{3} c^{6} - 6608 \, B a^{2} c^{7} - 12608 \, A a b c^{7}}{c^{7}}\right )} x + \frac{63 \, B b^{5} c^{4} - 568 \, B a b^{3} c^{5} - 112 \, A b^{4} c^{5} + 1392 \, B a^{2} b c^{6} + 960 \, A a b^{2} c^{6} + 18432 \, A a^{2} c^{7}}{c^{7}}\right )} x - \frac{315 \, B b^{6} c^{3} - 3164 \, B a b^{4} c^{4} - 560 \, A b^{5} c^{4} + 9552 \, B a^{2} b^{2} c^{5} + 5376 \, A a b^{3} c^{5} - 6720 \, B a^{3} c^{6} - 14592 \, A a^{2} b c^{6}}{c^{7}}\right )} x + \frac{945 \, B b^{7} c^{2} - 10500 \, B a b^{5} c^{3} - 1680 \, A b^{6} c^{3} + 37744 \, B a^{2} b^{3} c^{4} + 17920 \, A a b^{4} c^{4} - 42432 \, B a^{3} b c^{5} - 59136 \, A a^{2} b^{2} c^{5} + 49152 \, A a^{3} c^{6}}{c^{7}}\right )} + \frac{5 \,{\left (9 \, B b^{8} - 112 \, B a b^{6} c - 16 \, A b^{7} c + 480 \, B a^{2} b^{4} c^{2} + 192 \, A a b^{5} c^{2} - 768 \, B a^{3} b^{2} c^{3} - 768 \, A a^{2} b^{3} c^{3} + 256 \, B a^{4} c^{4} + 1024 \, A a^{3} b c^{4}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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