Optimal. Leaf size=641 \[ -\frac{4 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{4 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.949103, antiderivative size = 641, 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 = {738, 832, 843, 718, 424, 419} \[ \frac{4 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )}-\frac{4 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Rule 738
Rule 832
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{(d+e x)^{3/2} \left (-\frac{5}{2} e (b d-2 a e)-\frac{5}{2} e (2 c d-b e) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{c \left (b^2-4 a c\right )}-\frac{4 \int \frac{\sqrt{d+e x} \left (-\frac{5}{4} e \left (b^2 d e-16 a c d e+3 b \left (c d^2+a e^2\right )\right )-\frac{5}{2} e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=-\frac{2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{c \left (b^2-4 a c\right )}-\frac{8 \int \frac{\frac{5}{8} e \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )-\frac{5}{8} e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 c^2 \left (b^2-4 a c\right )}\\ &=-\frac{2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac{\left ((2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}-\frac{\left (8 \left (\frac{5}{8} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )+\frac{5}{8} e^2 \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 c^2 \left (b^2-4 a c\right ) e}\\ &=-\frac{2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac{\left (\sqrt{2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 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 )}{3 c^3 \sqrt{b^2-4 a c} \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} \left (\frac{5}{8} d e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )+\frac{5}{8} e^2 \left (4 b^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\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 )}{15 c^3 \sqrt{b^2-4 a c} e \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 (d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac{\sqrt{2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 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 )}{3 c^3 \sqrt{b^2-4 a c} \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} \left (c d^2-b d e+a e^2\right ) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2-5 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 )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 12.9804, size = 5433, normalized size = 8.48 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.404, size = 6486, normalized size = 10.1 \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 (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x + a\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 (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, 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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