Optimal. Leaf size=274 \[ \frac{d^{7/2} \left (b^2-4 a c\right )^{17/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{462 c^3 \sqrt{a+b x+c x^2}}+\frac{d^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c^2}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{110 c^2 d}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{385 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d} \]
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Rubi [A] time = 0.223885, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{d^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c^2}+\frac{d^{7/2} \left (b^2-4 a c\right )^{17/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{462 c^3 \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{110 c^2 d}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{385 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d} \]
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)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}-\frac{\left (b^2-4 a c\right ) \int (b d+2 c d x)^{7/2} \sqrt{a+b x+c x^2} \, dx}{10 c}\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}}{110 c^2 d}+\frac{(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac{\left (b^2-4 a c\right )^2 \int \frac{(b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \, dx}{220 c^2}\\ &=\frac{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{385 c^2}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}}{110 c^2 d}+\frac{(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac{\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac{(b d+2 c d x)^{3/2}}{\sqrt{a+b x+c x^2}} \, dx}{308 c^2}\\ &=\frac{\left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{231 c^2}+\frac{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{385 c^2}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}}{110 c^2 d}+\frac{(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac{\left (\left (b^2-4 a c\right )^4 d^4\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{924 c^2}\\ &=\frac{\left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{231 c^2}+\frac{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{385 c^2}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}}{110 c^2 d}+\frac{(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac{\left (\left (b^2-4 a c\right )^4 d^4 \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}{924 c^2 \sqrt{a+b x+c x^2}}\\ &=\frac{\left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{231 c^2}+\frac{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{385 c^2}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}}{110 c^2 d}+\frac{(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac{\left (\left (b^2-4 a c\right )^4 d^3 \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 )}{462 c^3 \sqrt{a+b x+c x^2}}\\ &=\frac{\left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{231 c^2}+\frac{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}{385 c^2}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}}{110 c^2 d}+\frac{(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac{\left (b^2-4 a c\right )^{17/4} d^{7/2} \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 )}{462 c^3 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.411386, size = 161, normalized size = 0.59 \[ \frac{4 \sqrt{a+x (b+c x)} (d (b+2 c x))^{7/2} \left (11 (b+2 c x)^2 (a+x (b+c x))^2-10 c \left (a-\frac{b^2}{4 c}\right ) \left (2 (a+x (b+c x))^2-\frac{\left (b^2-4 a c\right )^2 \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{16 c^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right )\right )}{165 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.271, size = 1057, normalized size = 3.9 \begin{align*} \text{result too large to display} \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{7}{2}}{\left (c x^{2} + b x + a\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 ({\left (8 \, c^{4} d^{3} x^{5} + 20 \, b c^{3} d^{3} x^{4} + a b^{3} d^{3} + 2 \,{\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{3} +{\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{2} +{\left (b^{4} + 6 \, a b^{2} c\right )} d^{3} x\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(-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.
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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{7}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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