Optimal. Leaf size=744 \[ -\frac{4 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{4 e \sqrt{a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.911197, antiderivative size = 744, 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 = {740, 834, 843, 718, 424, 419} \[ -\frac{4 e \sqrt{a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^3}-\frac{4 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 740
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} e \left (3 b c d-4 b^2 e+10 a c e\right )+\frac{3}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \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{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{4 \int \frac{\frac{1}{4} e \left (15 b^2 c d e-48 a c^2 d e-8 b^3 e^2-b c \left (3 c d^2-29 a e^2\right )\right )-\frac{1}{2} c e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \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{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 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 ) \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{8 \int \frac{\frac{1}{8} c 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{1}{8} c 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}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \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{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 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 ) \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{\left (2 c \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}+\frac{\left (c (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 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \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{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 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 ) \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}+\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 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3 \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} \left (3 c^2 d^2+2 b^2 e^2-c e (3 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 )}{3 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \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{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 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 ) \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}+\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 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3 \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 (3 c^2 d^2+2 b^2 e^2-c e (3 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 )}{3 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.3263, size = 5565, normalized size = 7.48 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.436, size = 12895, normalized size = 17.3 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c^{2} e^{3} x^{7} +{\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{6} +{\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{5} + a^{2} d^{3} +{\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{4} +{\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{3} +{\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{2} +{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{\frac{5}{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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]