Optimal. Leaf size=494 \[ -\frac{20 \sqrt{2} e \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{10 e \sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{2} e \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.414939, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {768, 738, 843, 718, 424, 419} \[ -\frac{20 \sqrt{2} e \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{10 e \sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{2} e \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 768
Rule 738
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac{1}{3} (5 e) \int \frac{(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{10 e \sqrt{d+e x} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{(10 e) \int \frac{-\frac{1}{2} e (b d-2 a e)-\frac{1}{2} e (2 c d-b e) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{10 e \sqrt{d+e x} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{(5 e (2 c d-b e)) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )}-\frac{\left (10 e \left (c d^2-b d e+a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{10 e \sqrt{d+e x} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\left (5 \sqrt{2} e (2 c d-b e) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}-\frac{\left (20 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 (d+e x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{10 e \sqrt{d+e x} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{2} e (2 c d-b e) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}-\frac{20 \sqrt{2} e \left (c d^2-b d e+a e^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 10.4069, size = 1150, normalized size = 2.33 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (10 d e x c^2-14 a e^2 c+5 b d e c-5 b e^2 x c+b^2 e^2\right )}{3 c \left (4 a c-b^2\right ) \left (c x^2+b x+a\right )}-\frac{2 \left (c d^2+2 c e x d-a e^2-b e^2 x\right )}{3 c \left (c x^2+b x+a\right )^2}\right ) \left (c x^2+b x+a\right )^3}{(a+x (b+c x))^{5/2}}+\frac{5 (d+e x)^{3/2} \left (-4 (b e-2 c d) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )+\frac{i \sqrt{2} (b e-2 c d) \left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left (\frac{2 d}{d+e x}-1\right ) e-2 c d \left (\frac{d}{d+e x}-1\right )+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left (\frac{d}{d+e x}-1\right )+b \left (e-\frac{2 d e}{d+e x}\right )+\sqrt{\left (b^2-4 a c\right ) e^2}}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt{d+e x}}-\frac{i \sqrt{2} \left (-b^2 e^2+4 a c e^2+b \sqrt{\left (b^2-4 a c\right ) e^2} e-2 c d \sqrt{\left (b^2-4 a c\right ) e^2}\right ) \sqrt{\frac{-\frac{2 a e^2}{d+e x}+b \left (\frac{2 d}{d+e x}-1\right ) e-2 c d \left (\frac{d}{d+e x}-1\right )+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} \sqrt{\frac{\frac{2 a e^2}{d+e x}+2 c d \left (\frac{d}{d+e x}-1\right )+b \left (e-\frac{2 d e}{d+e x}\right )+\sqrt{\left (b^2-4 a c\right ) e^2}}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right ),-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt{d+e x}}\right ) \left (c x^2+b x+a\right )^{5/2}}{6 c \left (b^2-4 a c\right ) \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} (a+x (b+c x))^{5/2} \sqrt{\frac{(d+e x)^2 \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.106, size = 5517, normalized size = 11.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, c x + b\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (2 \, c e^{2} x^{3} + b d^{2} +{\left (4 \, c d e + b e^{2}\right )} x^{2} + 2 \,{\left (c d^{2} + b d e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [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.
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