Optimal. Leaf size=992 \[ -\frac{2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (c^2 \left (24 C d^2-e (4 B d+A e)\right ) d^3-c e \left (b d \left (41 C d^2-6 B e d+A e^2\right )-a e \left (37 C d^2-7 B e d+7 A e^2\right )\right ) d+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B e d+2 A e^2\right )\right )+e \left (5 c^2 \left (6 C d^2-e (B d+A e)\right ) d^2+e^2 \left (\left (23 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (8 C d-B e) b+15 a^2 C e^2\right )-c e \left (5 b d \left (11 C d^2-2 B e d-A e^2\right )-a e \left (53 C d^2-13 B e d+3 A e^2\right )\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{15 e^3 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (2 c^2 \left (24 C d^2-e (4 B d+A e)\right ) d^2+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 a^2 C e^2\right )-c e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 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^4 \left (c d^2-b e d+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{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (2 d \left (24 C d^2-e (4 B d+A e)\right ) c^2+e \left (10 a e (5 C d-B e)-b \left (64 C d^2-9 B e d-A e^2\right )\right ) c+15 b C e^2 (b d-a e)\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 c e^4 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]
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Rubi [A] time = 1.91633, antiderivative size = 989, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1650, 810, 843, 718, 424, 419} \[ -\frac{2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (\left (24 C d^5-d^3 e (4 B d+A e)\right ) c^2-d e \left (b d \left (41 C d^2-6 B e d+A e^2\right )-a e \left (37 C d^2-7 B e d+7 A e^2\right )\right ) c+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B e d+2 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}-5 c^2 (B d+A e) d^2-5 b c \left (11 C d^2-e (2 B d+A e)\right ) d+15 a^2 C e^3-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{15 e^3 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (\left (48 C d^4-2 d^2 e (4 B d+A e)\right ) c^2-e \left (b d \left (88 C d^2-13 B e d-2 A e^2\right )-2 a e \left (43 C d^2-8 B e d+3 A e^2\right )\right ) c+e^2 \left (\left (38 C d^2-3 B e d-2 A e^2\right ) b^2-5 a e (14 C d-B e) b+30 a^2 C e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^4 \left (c d^2-b e d+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (\left (48 C d^3-2 d e (4 B d+A e)\right ) c^2-e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\right ) c+15 b C e^2 (b d-a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 c e^4 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 810
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{7/2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \int \frac{\left (-\frac{3 b C d^2-b e (3 B d+2 A e)+5 e (A c d-a C d+a B e)}{2 e}+\frac{1}{2} \left (B c d+5 b C d-\frac{6 c C d^2}{e}-A c e-5 a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^{5/2}} \, dx}{5 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (c^2 \left (24 C d^5-d^3 e (4 B d+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{4 \int \frac{\frac{15 b^3 C d^2 e^2-b^2 d e \left (41 c C d^2+30 a C e^2-c e (6 B d-A e)\right )-2 a c e \left (6 c C d^3-c d e (B d+4 A e)+5 a e^2 (2 C d-B e)\right )+b \left (15 a^2 C e^4+c^2 \left (24 C d^4-d^2 e (4 B d+A e)\right )+a c e^2 \left (59 C d^2-e (14 B d+A e)\right )\right )}{4 e}+\frac{c \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 A e^2\right )\right )\right ) x}{4 e}}{\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{2 \left (c^2 \left (24 C d^5-d^3 e (4 B d+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{\left (15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (4 B d+A e)\right )-c e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 e^4 \left (c d^2-b d e+a e^2\right )}+\frac{\left (c \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 A e^2\right )\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{15 e^4 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (c^2 \left (24 C d^5-d^3 e (4 B d+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 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^4 \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} \sqrt{b^2-4 a c} \left (15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (4 B d+A e)\right )-c e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B 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 c e^4 \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 (c^2 \left (24 C d^5-d^3 e (4 B d+A e)\right )+e^2 \left (15 b^2 C d^3+5 a^2 e^2 (C d+B e)-a b e \left (22 C d^2+3 B d e+2 A e^2\right )\right )-c d e \left (b d \left (41 C d^2-6 B d e+A e^2\right )-a e \left (37 C d^2-7 B d e+7 A e^2\right )\right )+e^2 \left (\frac{30 c^2 C d^4}{e}+15 a^2 C e^3-5 c^2 d^2 (B d+A e)-5 a b e^2 (8 C d-B e)+a c e \left (53 C d^2-e (13 B d-3 A e)\right )-5 b c d \left (11 C d^2-e (2 B d+A e)\right )+b^2 e \left (23 C d^2-e (3 B d+2 A e)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (48 C d^4-2 d^2 e (4 B d+A e)\right )+e^2 \left (30 a^2 C e^2-5 a b e (14 C d-B e)+b^2 \left (38 C d^2-3 B d e-2 A e^2\right )\right )-c e \left (b d \left (88 C d^2-13 B d e-2 A e^2\right )-2 a e \left (43 C d^2-8 B d e+3 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^4 \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} \sqrt{b^2-4 a c} \left (15 b C e^2 (b d-a e)+c^2 \left (48 C d^3-2 d e (4 B d+A e)\right )-c e \left (64 b C d^2-b e (9 B d+A e)-10 a e (5 C d-B 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 c e^4 \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 = 14.864, size = 12997, normalized size = 13.1 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.566, size = 48427, normalized size = 48.8 \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 (C x^{2} + B x + A\right )} \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}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, 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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