Optimal. Leaf size=180 \[ -\frac{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{21 c^2 \sqrt{a+b x+c x^2}}-\frac{2 d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{21 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d} \]
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Rubi [A] time = 0.148845, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {685, 692, 691, 689, 221} \[ -\frac{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c^2 \sqrt{a+b x+c x^2}}-\frac{2 d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{21 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d} \]
Antiderivative was successfully verified.
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Rule 685
Rule 692
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2} \, dx &=\frac{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{7 c d}-\frac{\left (b^2-4 a c\right ) \int \frac{(b d+2 c d x)^{3/2}}{\sqrt{a+b x+c x^2}} \, dx}{14 c}\\ &=-\frac{2 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{21 c}+\frac{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{7 c d}-\frac{\left (\left (b^2-4 a c\right )^2 d^2\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{42 c}\\ &=-\frac{2 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{21 c}+\frac{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{7 c d}-\frac{\left (\left (b^2-4 a c\right )^2 d^2 \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}{42 c \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{21 c}+\frac{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{7 c d}-\frac{\left (\left (b^2-4 a c\right )^2 d \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 )}{21 c^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{21 c}+\frac{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{7 c d}-\frac{\left (b^2-4 a c\right )^{9/4} d^{3/2} \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 )}{21 c^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.137379, size = 110, normalized size = 0.61 \[ \frac{1}{14} d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \left (\frac{\left (b^2-4 a c\right ) \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}+8 (a+x (b+c x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.207, size = 564, normalized size = 3.1 \begin{align*} -{\frac{d}{42\,{c}^{2} \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\,{x}^{5}{c}^{5}+16\,\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 EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{a}^{2}{c}^{2}-8\,\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 EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}a{b}^{2}c+\sqrt{{ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{ \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{4}-120\,{x}^{4}b{c}^{4}-80\,{x}^{3}a{c}^{4}-100\,{x}^{3}{b}^{2}{c}^{3}-120\,{x}^{2}ab{c}^{3}-30\,{x}^{2}{b}^{3}{c}^{2}-32\,x{a}^{2}{c}^{3}-44\,xa{b}^{2}{c}^{2}-2\,x{b}^{4}c-16\,{a}^{2}b{c}^{2}-2\,a{b}^{3}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{\left (2 \, c d x + b d\right )}^{\frac{3}{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 ({\left (2 \, c d x + b d\right )}^{\frac{3}{2}} \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 \left (d \left (b + 2 c x\right )\right )^{\frac{3}{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{\left (2 \, c d x + b d\right )}^{\frac{3}{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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