Optimal. Leaf size=231 \[ -\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{4}{5} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2} \]
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Rubi [A] time = 0.215393, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {692, 691, 690, 307, 221, 1199, 424} \[ -\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{6 d^{5/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c \sqrt{a+b x+c x^2}}+\frac{4}{5} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}} \, dx &=\frac{4}{5} d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{1}{5} \left (3 \left (b^2-4 a c\right ) d^2\right ) \int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx\\ &=\frac{4}{5} d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{\left (3 \left (b^2-4 a c\right ) d^2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac{\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}{5 \sqrt{a+b x+c x^2}}\\ &=\frac{4}{5} d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{\left (6 \left (b^2-4 a c\right ) d \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{5 c \sqrt{a+b x+c x^2}}\\ &=\frac{4}{5} d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}-\frac{\left (6 \left (b^2-4 a c\right )^{3/2} d^2 \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 )}{5 c \sqrt{a+b x+c x^2}}+\frac{\left (6 \left (b^2-4 a c\right )^{3/2} d^2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{\sqrt{b^2-4 a c} d}}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{5 c \sqrt{a+b x+c x^2}}\\ &=\frac{4}{5} d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}-\frac{6 \left (b^2-4 a c\right )^{7/4} d^{5/2} \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 )}{5 c \sqrt{a+b x+c x^2}}+\frac{\left (6 \left (b^2-4 a c\right )^{3/2} d^2 \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{x^2}{\sqrt{b^2-4 a c} d}}}{\sqrt{1-\frac{x^2}{\sqrt{b^2-4 a c} d}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{5 c \sqrt{a+b x+c x^2}}\\ &=\frac{4}{5} d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{6 \left (b^2-4 a c\right )^{7/4} d^{5/2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{5 c \sqrt{a+b x+c x^2}}-\frac{6 \left (b^2-4 a c\right )^{7/4} d^{5/2} \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 )}{5 c \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.126672, size = 111, normalized size = 0.48 \[ \frac{2 d (d (b+2 c x))^{3/2} \left (\left (b^2-4 a c\right ) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )+2 c (a+x (b+c x))\right )}{5 c \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.208, size = 496, normalized size = 2.2 \begin{align*} -{\frac{{d}^{2}}{5\,c \left ( 2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 48\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{c}^{2}-24\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{2}c+3\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{4}-16\,{c}^{4}{x}^{4}-32\,b{c}^{3}{x}^{3}-16\,{x}^{2}a{c}^{3}-20\,{x}^{2}{b}^{2}{c}^{2}-16\,ba{c}^{2}x-4\,{b}^{3}cx-4\,ac{b}^{2} \right ) } \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 (2 \, c d x + b d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x + a}}\,{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 (4 \, c^{2} d^{2} x^{2} + 4 \, b c d^{2} x + b^{2} d^{2}\right )} \sqrt{2 \, c d x + b d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}}}{\sqrt{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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