Optimal. Leaf size=705 \[ \frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{2 e \sqrt{a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]
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Rubi [A] time = 0.846731, antiderivative size = 705, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {822, 834, 843, 718, 424, 419} \[ \frac{2 e \sqrt{a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} e \left (b^2 (B d-2 A e)-6 a c (B d-A e)+b (A c d+a B e)\right )-\frac{1}{2} c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 e \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}+\frac{4 \int \frac{-\frac{1}{4} c e \left (b^2 d (2 B d-A e)-b \left (A c d^2+2 a B d e+a A e^2\right )-2 a \left (3 B c d^2-4 A c d e-a B e^2\right )\right )-\frac{1}{4} c e \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 e \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}+\frac{(c (b B d-2 A c d+A b e-2 a B e)) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}-\frac{\left (c \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 e \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}-\frac{\left (\sqrt{2} \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \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 )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \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 (2 \sqrt{2} (b B d-2 A c d+A b e-2 a B e) \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 )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 e \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}-\frac{\sqrt{2} \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \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 )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \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{2 \sqrt{2} (b B d-2 A c d+A b e-2 a B e) \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 )}{\sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 12.0921, size = 1267, normalized size = 1.8 \[ \frac{\sqrt{d+e x} \left (c x^2+b x+a\right )^2 \left (\frac{2 \left (A e^2 b^3-a B e^2 b^2-2 A c d e b^2+A c e^2 x b^2+A c^2 d^2 b-3 a A c e^2 b+2 a B c d e b-B c^2 d^2 x b-a B c e^2 x b-2 A c^2 d e x b-2 a B c^2 d^2+2 a^2 B c e^2+4 a A c^2 d e+2 A c^3 d^2 x-2 a A c^2 e^2 x+4 a B c^2 d e x\right )}{\left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )}-\frac{2 e^2 (A e-B d)}{\left (c d^2-b e d+a e^2\right )^2 (d+e x)}\right )}{(a+x (b+c x))^{3/2}}-\frac{2 (d+e x)^{3/2} \left (c x^2+b x+a\right )^{3/2} \left (-\left (e (B d-2 A e) b^2+\left (B c d^2+2 A c e d+a B e^2\right ) b-2 c \left (A c d^2+4 a B e d-3 a A e^2\right )\right ) \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{1-\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+1} \left (\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \left (e (2 A e-B d) b^2-\left (B c d^2+2 A c e d+a B e^2\right ) b+2 c \left (A c d^2+4 a B e d-3 a A e^2\right )\right ) 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 )+\left (e^2 (2 A e-B d) b^3-e \left (2 A e \left (2 c d+\sqrt{\left (b^2-4 a c\right ) e^2}\right )+B \left (-3 c d^2-\sqrt{\left (b^2-4 a c\right ) e^2} d+a e^2\right )\right ) b^2+\left (a \left (4 c d B+\sqrt{\left (b^2-4 a c\right ) e^2} B-8 A c e\right ) e^2+c d \sqrt{\left (b^2-4 a c\right ) e^2} (B d+2 A e)\right ) b-2 c \left (-2 a^2 B e^3+a \left (6 B c d^2-8 A c e d+4 B \sqrt{\left (b^2-4 a c\right ) e^2} d-3 A e \sqrt{\left (b^2-4 a c\right ) e^2}\right ) e+A c d^2 \sqrt{\left (b^2-4 a c\right ) e^2}\right )\right ) \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 )\right )}{2 \sqrt{2} \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}}} \sqrt{d+e x}}\right )}{\left (4 a c-b^2\right ) e \left (c d^2-b e d+a e^2\right )^2 (a+x (b+c x))^{3/2} \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 = 8357, normalized size = 11.9 \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{B x + A}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{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{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )} \sqrt{e x + d}}{c^{2} e^{2} x^{6} + 2 \,{\left (c^{2} d e + b c e^{2}\right )} x^{5} +{\left (c^{2} d^{2} + 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{4} + a^{2} d^{2} + 2 \,{\left (b c d^{2} + a b e^{2} +{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{3} +{\left (4 \, a b d e + a^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{2} + 2 \,{\left (a b d^{2} + a^{2} d e\right )} x}, 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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