Optimal. Leaf size=944 \[ -\frac{2 \sqrt{c x^2+b x+a} \left (C d^2-e (B d-A e)\right )}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (2 C d^2+e (3 B d-23 A e)\right ) d^2-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )-c e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d \left (2 C d^2+e (3 B d-8 A e)\right )+e \left (5 a e (2 C d-B e)-b \left (6 C d^2-B e d-4 A e^2\right )\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}+\frac{2 \left (c^2 \left (2 C d^2+e (3 B d-23 A e)\right ) d^2-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )-c e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^3 \sqrt{d+e x}}+\frac{2 \left (c d \left (2 C d^2+e (3 B d-8 A e)\right )+e \left (5 a e (2 C d-B e)-b \left (6 C d^2-B e d-4 A e^2\right )\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}} \]
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Rubi [A] time = 2.25575, antiderivative size = 942, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1650, 834, 843, 718, 424, 419} \[ -\frac{2 \sqrt{c x^2+b x+a} \left (C d^2-e (B d-A e)\right )}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (\left (2 C d^4+e (3 B d-23 A e) d^2\right ) c^2-e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right ) c-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (2 c C d^3+c e (3 B d-8 A e) d+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}+\frac{2 \left (\left (2 C d^4+e (3 B d-23 A e) d^2\right ) c^2-e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right ) c-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^3 \sqrt{d+e x}}+\frac{2 \left (2 c C d^3+c e (3 B d-8 A e) d+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \int \frac{-\frac{b C d^2-b e (B d+4 A e)+5 e (A c d-a C d+a B e)}{2 e}-\frac{1}{2} \left (3 B c d-5 b C d+\frac{2 c C d^2}{e}-3 A c e+5 a C e\right ) x}{(d+e x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{5 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{4 \int \frac{\frac{b^2 e \left (3 C d^2+2 e (B d+4 A e)\right )+b \left (c C d^3-c d e (6 B d+19 A e)-10 a e^2 (C d+B e)\right )+3 e \left (A c \left (5 c d^2-3 a e^2\right )+a \left (5 a C e^2-c d (3 C d-8 B e)\right )\right )}{4 e}+\frac{c \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) x}{4 e}}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{8 \int \frac{-\frac{c \left (b^2 d e \left (9 C d^2+e (B d+4 A e)\right )-b \left (c C d^4+26 a C d^2 e^2+4 a e^3 (B d-A e)+c d^2 e (9 B d+11 A e)\right )+e \left (A c d \left (15 c d^2-17 a e^2\right )-a \left (c d^2 (7 C d-27 B e)-5 a e^2 (5 C d-B e)\right )\right )\right )}{8 e}+\frac{c \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) x}{8 e}}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}+\frac{\left (c \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^2}-\frac{\left (c \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^3 \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} \sqrt{b^2-4 a c} \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^3 \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} \sqrt{b^2-4 a c} \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 15.2817, size = 12295, normalized size = 13.02 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.628, size = 46695, normalized size = 49.5 \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{C x^{2} + B x + A}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{7}{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 (C x^{2} + B x + A\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c e^{4} x^{6} +{\left (4 \, c d e^{3} + b e^{4}\right )} x^{5} + a d^{4} +{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{4} + 2 \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{3} +{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{2} +{\left (b d^{4} + 4 \, a d^{3} 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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