Optimal. Leaf size=316 \[ -\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{12 c^3 d^3}+\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}} \]
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Rubi [A] time = 0.292052, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {684, 685, 691, 690, 307, 221, 1199, 424} \[ -\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{12 c^3 d^3}-\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}} \]
Antiderivative was successfully verified.
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Rule 684
Rule 685
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}+\frac{5 \int \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx}{2 c d^2}\\ &=\frac{5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}-\frac{\left (5 \left (b^2-4 a c\right )\right ) \int \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2} \, dx}{12 c^2 d^2}\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}{12 c^3 d^3}+\frac{5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}+\frac{\left (b^2-4 a c\right )^2 \int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx}{24 c^3 d^2}\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}{12 c^3 d^3}+\frac{5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}+\frac{\left (\left (b^2-4 a c\right )^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}{24 c^3 d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}{12 c^3 d^3}+\frac{5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}+\frac{\left (\left (b^2-4 a c\right )^2 \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 )}{12 c^4 d^3 \sqrt{a+b x+c x^2}}\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}{12 c^3 d^3}+\frac{5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}-\frac{\left (\left (b^2-4 a c\right )^{5/2} \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 )}{12 c^4 d^2 \sqrt{a+b x+c x^2}}+\frac{\left (\left (b^2-4 a c\right )^{5/2} \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 )}{12 c^4 d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}{12 c^3 d^3}+\frac{5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}-\frac{\left (b^2-4 a c\right )^{11/4} \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 d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left (\left (b^2-4 a c\right )^{5/2} \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 )}{12 c^4 d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}{12 c^3 d^3}+\frac{5 (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}}+\frac{\left (b^2-4 a c\right )^{11/4} \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 d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right )^{11/4} \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 d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0536455, size = 101, normalized size = 0.32 \[ -\frac{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \, _2F_1\left (-\frac{5}{2},-\frac{1}{4};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.227, size = 700, normalized size = 2.2 \begin{align*}{\frac{1}{72\,{c}^{4}{d}^{2} \left ( 2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab \right ) }\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 8\,{x}^{6}{c}^{6}+24\,{x}^{5}b{c}^{5}+192\,\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}^{3}{c}^{3}-144\,\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}{b}^{2}{c}^{2}+36\,\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}^{4}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}^{6}+40\,{x}^{4}a{c}^{5}+20\,{x}^{4}{b}^{2}{c}^{4}+80\,{x}^{3}ab{c}^{4}-40\,{x}^{2}{a}^{2}{c}^{4}+80\,{x}^{2}a{b}^{2}{c}^{3}-10\,{x}^{2}{b}^{4}{c}^{2}-40\,x{a}^{2}b{c}^{3}+40\,xa{b}^{3}{c}^{2}-6\,x{b}^{5}c-72\,{a}^{3}{c}^{3}+44\,{a}^{2}{b}^{2}{c}^{2}-6\,a{b}^{4}c \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 (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\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 (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}}{4 \, c^{2} d^{2} x^{2} + 4 \, b c d^{2} x + b^{2} d^{2}}, 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 (a + b x + c x^{2}\right )^{\frac{5}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{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 (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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