Optimal. Leaf size=701 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{15 e^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}+\frac{2 c \sqrt{a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{15 e^4 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^5 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]
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Rubi [A] time = 0.754707, antiderivative size = 701, 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 = {810, 812, 843, 718, 424, 419} \[ -\frac{2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}+\frac{2 c \sqrt{a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{15 e^4 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^5 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 812
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx &=-\frac{2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \int \frac{\left (-\frac{1}{2} c \left (16 b c d^2-13 b^2 d e-12 a c d e+16 a b e^2\right )-\frac{1}{2} c \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{4 \int \frac{-\frac{1}{4} c \left (51 b^3 d e^2-8 a c e \left (8 c d^2+5 a e^2\right )+4 b c d \left (32 c d^2+45 a e^2\right )-2 b^2 \left (88 c d^2 e+27 a e^3\right )\right )-\frac{1}{4} c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{\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{2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{15 e^5 \left (c d^2-b d e+a e^2\right )}+\frac{\left (2 c \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 e^5}\\ &=\frac{2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \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} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\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^5 \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{a+b x+c x^2}}+\frac{\left (4 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\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^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{15 e^4 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 \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} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\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^5 \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{a+b x+c x^2}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 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 (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^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 6.95768, size = 5450, normalized size = 7.77 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.108, size = 19271, normalized size = 27.5 \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 )}^{\frac{3}{2}}{\left (2 \, c x + b\right )}}{{\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 (2 \, c^{2} x^{3} + 3 \, b c x^{2} + a b +{\left (b^{2} + 2 \, a c\right )} x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{7}{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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