Optimal. Leaf size=600 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{105 c^3}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{12 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.848873, antiderivative size = 600, 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, 832, 843, 718, 424, 419} \[ \frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{105 c^3}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{12 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 742
Rule 832
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\sqrt{a+b x+c x^2}} \, dx &=\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 \int \frac{(d+e x)^{3/2} \left (\frac{1}{2} \left (7 c d^2-e (b d+5 a e)\right )+3 e (2 c d-b e) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{7 c}\\ &=\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{4 \int \frac{\sqrt{d+e x} \left (\frac{1}{4} \left (35 c^2 d^3+6 b e^2 (b d+3 a e)-c d e (17 b d+61 a e)\right )+\frac{1}{4} e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{35 c^2}\\ &=\frac{2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{8 \int \frac{\frac{1}{8} \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )+e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3}\\ &=\frac{2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{\left (8 (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^3}+\frac{\left (8 \left (-d e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )+\frac{1}{8} e \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 e}\\ &=\frac{2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{\left (8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 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 )}{105 c^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 (-d e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )+\frac{1}{8} e \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\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 )}{105 c^4 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 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 )}{105 c^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 (71 c^2 d^2-71 b c d e+24 b^2 e^2-25 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 )}{105 c^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.1642, size = 5340, normalized size = 8.9 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.341, size = 6947, normalized size = 11.6 \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{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x + a}}\,{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{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[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]