Optimal. Leaf size=325 \[ \frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{112 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}-\frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{28 c}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}} \]
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Rubi [A] time = 0.283973, antiderivative size = 325, 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 = {687, 693, 691, 690, 307, 221, 1199, 424} \[ \frac{112 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}+\frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{28 c}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}-\frac{(14 c) \int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac{28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}+\frac{\left (28 c^2\right ) \int \frac{1}{(b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac{28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}+\frac{112 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt{b d+2 c d x}}-\frac{\left (28 c^2\right ) \int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac{28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}+\frac{112 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt{b d+2 c d x}}-\frac{\left (28 c^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}{\left (b^2-4 a c\right )^3 d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac{28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}+\frac{112 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt{b d+2 c d x}}-\frac{\left (56 c \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 )}{\left (b^2-4 a c\right )^3 d^3 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac{28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}+\frac{112 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt{b d+2 c d x}}+\frac{\left (56 c \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt{a+b x+c x^2}}-\frac{\left (56 c \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac{28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}+\frac{112 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt{b d+2 c d x}}+\frac{56 c \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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left (56 c \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac{28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}+\frac{112 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt{b d+2 c d x}}-\frac{56 c \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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt{a+b x+c x^2}}+\frac{56 c \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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0577227, size = 97, normalized size = 0.3 \[ -\frac{32 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (-\frac{1}{4},\frac{5}{2};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.239, size = 877, normalized size = 2.7 \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 \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{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{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{4 \, c^{5} d^{2} x^{8} + 16 \, b c^{4} d^{2} x^{7} +{\left (25 \, b^{2} c^{3} + 12 \, a c^{4}\right )} d^{2} x^{6} +{\left (19 \, b^{3} c^{2} + 36 \, a b c^{3}\right )} d^{2} x^{5} + a^{3} b^{2} d^{2} +{\left (7 \, b^{4} c + 39 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{2} x^{4} +{\left (b^{5} + 18 \, a b^{3} c + 24 \, a^{2} b c^{2}\right )} d^{2} x^{3} +{\left (3 \, a b^{4} + 15 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{2} x^{2} +{\left (3 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{2} x}, 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{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{5}{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{1}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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