Optimal. Leaf size=305 \[ \frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{17556 c^4 d^{21/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 d^9 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.262961, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {684, 693, 691, 689, 221} \[ \frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 d^9 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}+\frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{17556 c^4 d^{21/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 684
Rule 693
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{21/2}} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{5 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{17/2}} \, dx}{38 c d^2}\\ &=-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx}{76 c^2 d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{\int \frac{1}{(b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}} \, dx}{1672 c^3 d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{5 \int \frac{1}{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{11704 c^3 \left (b^2-4 a c\right ) d^8}\\ &=-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{5 \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{35112 c^3 \left (b^2-4 a c\right )^2 d^{10}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{\left (5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{-\frac{a c}{b^2-4 a c}-\frac{b c x}{b^2-4 a c}-\frac{c^2 x^2}{b^2-4 a c}}} \, dx}{35112 c^3 \left (b^2-4 a c\right )^2 d^{10} \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{\left (5 \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-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{17556 c^4 \left (b^2-4 a c\right )^2 d^{11} \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{836 c^3 d^5 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{2926 c^3 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{7/2}}+\frac{5 \sqrt{a+b x+c x^2}}{8778 c^3 \left (b^2-4 a c\right )^2 d^9 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{114 c^2 d^3 (b d+2 c d x)^{15/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{19 c d (b d+2 c d x)^{19/2}}+\frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{17556 c^4 \left (b^2-4 a c\right )^{7/4} d^{21/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0846049, size = 109, normalized size = 0.36 \[ -\frac{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \, _2F_1\left (-\frac{19}{4},-\frac{5}{2};-\frac{15}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{608 c^3 d^9 (b+2 c x)^8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.287, size = 1843, normalized size = 6. \begin{align*} \text{result 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 (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{21}{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{{\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{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{2048 \, c^{11} d^{11} x^{11} + 11264 \, b c^{10} d^{11} x^{10} + 28160 \, b^{2} c^{9} d^{11} x^{9} + 42240 \, b^{3} c^{8} d^{11} x^{8} + 42240 \, b^{4} c^{7} d^{11} x^{7} + 29568 \, b^{5} c^{6} d^{11} x^{6} + 14784 \, b^{6} c^{5} d^{11} x^{5} + 5280 \, b^{7} c^{4} d^{11} x^{4} + 1320 \, b^{8} c^{3} d^{11} x^{3} + 220 \, b^{9} c^{2} d^{11} x^{2} + 22 \, b^{10} c d^{11} x + b^{11} d^{11}}, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{21}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]