Optimal. Leaf size=724 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{105 c^3 e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e \left (a e (63 B e+82 C d)+b \left (70 A e^2+91 B d e+12 C d^2\right )\right )-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 d \left (6 C d^2-7 e (20 A e+3 B d)\right )+48 b^3 C e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e} \]
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Rubi [A] time = 1.77854, antiderivative size = 724, 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 = {1653, 832, 843, 718, 424, 419} \[ \frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{105 c^3 e}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e \left (a e (63 B e+82 C d)+b \left (70 A e^2+91 B d e+12 C d^2\right )\right )-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 \left (6 C d^3-7 d e (20 A e+3 B d)\right )+48 b^3 C e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 832
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt{a+b x+c x^2}} \, dx &=\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}+\frac{2 \int \frac{(d+e x)^{3/2} \left (-\frac{1}{2} e (b C d-7 A c e+5 a C e)-\frac{1}{2} e (2 c C d-7 B c e+6 b C e) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{7 c e^2}\\ &=-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}+\frac{4 \int \frac{\sqrt{d+e x} \left (\frac{1}{4} e \left (6 b^2 C d e+18 a b C e^2-b c d (3 C d+7 B e)+c e (35 A c d-19 a C d-21 a B e)\right )+\frac{1}{4} e \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{35 c^2 e^2}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}+\frac{8 \int \frac{-\frac{1}{8} e \left (24 b^3 C d e^2+b c \left (3 c C d^3+7 c d e (6 B d+5 A e)-2 a e^2 (47 C d+14 B e)\right )+b^2 \left (24 a C e^3-c d e (33 C d+28 B e)\right )-c e \left (35 A c \left (3 c d^2-a e^2\right )-a \left (51 c C d^2+84 B c d e-25 a C e^2\right )\right )\right )-\frac{1}{8} e \left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 A e^2\right )\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 e^2}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}-\frac{\left (\left (c d^2-b d e+a e^2\right ) \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 e^2}-\frac{\left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 A e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^3 e^2}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 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 )}{105 c^4 e^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 (c d^2-b d e+a e^2\right ) \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 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 )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 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 )}{105 c^4 e^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 (c d^2-b d e+a e^2\right ) \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 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 )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 14.5728, size = 9972, normalized size = 13.77 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.438, size = 14084, normalized size = 19.5 \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{{\left (C x^{2} + B x + A\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c x^{2} + b x + a}}\,{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 e x^{3} +{\left (C d + B e\right )} x^{2} + A d +{\left (B d + A e\right )} x\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \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 \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A + B x + C x^{2}\right )}{\sqrt{a + b x + c x^{2}}}\, dx \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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