Optimal. Leaf size=267 \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{384 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{6144 c^4}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a c C+32 A c^2+9 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c} \]
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Rubi [A] time = 0.23808, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{384 c^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{6144 c^4}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c C+32 A c^2+9 b^2 C\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a c C+32 A c^2+9 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c} \]
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 )^{5/2} \left (A+C x^2\right ) \, dx &=\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac{\int \left (8 A c-a C-\frac{9 b C x}{2}\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac{\left (\frac{9 b^2 C}{2}+2 c (8 A c-a C)\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac{\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c 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 (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c 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 (32 A c^2+9 b^2 C-4 a c 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 (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c 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 (32 A c^2+9 b^2 C-4 a c 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 (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 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 )}{16384 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c 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 (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac{9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac{C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac{5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 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 )}{32768 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.851162, size = 344, normalized size = 1.29 \[ \frac{-\frac{1120 A \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )}{c^{5/2}}+57344 A (b+2 c x) (a+x (b+c x))^{5/2}+\frac{7 C \left (9 b^2-4 a c\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{c^{9/2}}-\frac{55296 b C (a+x (b+c x))^{7/2}}{c}+86016 C x (a+x (b+c x))^{7/2}}{688128 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 997, normalized size = 3.7 \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 = 2.88837, size = 2342, normalized size = 8.77 \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 \left (A + C x^{2}\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 = 1.25004, size = 651, normalized size = 2.44 \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 \, C c^{2} x + 33 \, C b c\right )} x + \frac{243 \, C b^{2} c^{7} + 476 \, C a c^{8} + 224 \, A c^{9}}{c^{7}}\right )} x + \frac{3 \, C b^{3} c^{6} + 1228 \, C a b c^{7} + 1120 \, A b c^{8}}{c^{7}}\right )} x - \frac{27 \, C b^{4} c^{5} - 216 \, C a b^{2} c^{6} - 6608 \, C a^{2} c^{7} - 6048 \, A b^{2} c^{7} - 11648 \, A a c^{8}}{c^{7}}\right )} x + \frac{63 \, C b^{5} c^{4} - 568 \, C a b^{3} c^{5} + 1392 \, C a^{2} b c^{6} + 224 \, A b^{3} c^{6} + 34944 \, A a b c^{7}}{c^{7}}\right )} x - \frac{315 \, C b^{6} c^{3} - 3164 \, C a b^{4} c^{4} + 9552 \, C a^{2} b^{2} c^{5} + 1120 \, A b^{4} c^{5} - 6720 \, C a^{3} c^{6} - 10752 \, A a b^{2} c^{6} - 118272 \, A a^{2} c^{7}}{c^{7}}\right )} x + \frac{945 \, C b^{7} c^{2} - 10500 \, C a b^{5} c^{3} + 37744 \, C a^{2} b^{3} c^{4} + 3360 \, A b^{5} c^{4} - 42432 \, C a^{3} b c^{5} - 35840 \, A a b^{3} c^{5} + 118272 \, A a^{2} b c^{6}}{c^{7}}\right )} + \frac{5 \,{\left (9 \, C b^{8} - 112 \, C a b^{6} c + 480 \, C a^{2} b^{4} c^{2} + 32 \, A b^{6} c^{2} - 768 \, C a^{3} b^{2} c^{3} - 384 \, A a b^{4} c^{3} + 256 \, C a^{4} c^{4} + 1536 \, A a^{2} b^{2} c^{4} - 2048 \, A a^{3} c^{5}\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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