Optimal. Leaf size=816 \[ \frac{2 e \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{1155 c^3 e^3}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]
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Rubi [A] time = 2.83596, antiderivative size = 816, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {742, 814, 843, 718, 424, 419} \[ \frac{2 e \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{1155 c^3 e^3}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]
Antiderivative was successfully verified.
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Rule 742
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac{2 \int \frac{\left (\frac{1}{2} \left (11 c d^2-e (5 b d+a e)\right )+3 e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 c}\\ &=\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{4 \int \frac{\left (\frac{3}{4} e \left (b c^2 d^3+13 b^2 c d^2 e-58 a c^2 d^2 e-6 b^3 d e^2+27 a b c d e^2-2 a b^2 e^3+6 a^2 c e^3\right )+\frac{3}{4} e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac{8 \int \frac{-\frac{3}{8} e \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac{1}{2} b d (4 c d-b e)-a e \left (c d+\frac{b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )-3 e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{1155 c^3 e^4}+\frac{\left (8 \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac{3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac{1}{2} b d (4 c d-b e)-a e \left (c d+\frac{b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3465 c^3 e^5}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 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 )}{1155 c^4 e^4 \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 (16 \sqrt{2} \sqrt{b^2-4 a c} \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac{3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac{1}{2} b d (4 c d-b e)-a e \left (c d+\frac{b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\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 )}{3465 c^4 e^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 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 )}{1155 c^4 e^4 \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 (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+84 a c^3 d^2 e^2+13 b^3 c d e^3-84 a b c^2 d e^3-8 b^4 e^4+51 a b^2 c e^4-60 a^2 c^2 e^4\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 )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.9613, size = 10848, normalized size = 13.29 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.407, size = 11933, normalized size = 14.6 \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{\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 ({\left (c e x^{3} +{\left (c d + b e\right )} x^{2} + a d +{\left (b d + a e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}, 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 \left (d + e x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{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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