Optimal. Leaf size=622 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{21 c e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{21 e^5}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{10 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.805415, antiderivative size = 622, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {732, 812, 814, 843, 718, 424, 419} \[ \frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{21 e^5}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{21 c e^6 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{10 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 732
Rule 812
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac{5 \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e}\\ &=\frac{10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac{10 \int \frac{\left (\frac{1}{2} \left (16 b c d-7 b^2 e-4 a c e\right )+8 c (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{7 e^3}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{21 e^5}+\frac{10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac{4 \int \frac{-\frac{1}{4} c \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\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}{21 c e^5}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{21 e^5}+\frac{10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac{\left ((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}{21 e^6}+\frac{\left (4 \left (-\frac{1}{4} c e \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )+\frac{1}{4} c d (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 )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{21 c e^6}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{21 e^5}+\frac{10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\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 )}{21 c e^6 \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 (8 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{1}{4} c e \left (128 b c^2 d^3-176 b^2 c d^2 e-64 a c^2 d^2 e+51 b^3 d e^2+180 a b c d e^2-54 a b^2 e^3-40 a^2 c e^3\right )+\frac{1}{4} c d (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 ) \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 )}{21 c^2 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{21 e^5}+\frac{10 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\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 )}{21 c e^6 \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 (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-128 b c d e+27 b^2 e^2+20 a c e^2\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 )}{21 c e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.5183, size = 5407, normalized size = 8.69 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.398, size = 12847, normalized size = 20.7 \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{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{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^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, 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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