Optimal. Leaf size=325 \[ -\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}+\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}} \]
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Rubi [A] time = 0.293854, 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} \[ -\frac{168 c \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}-\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}+\frac{84 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}} \]
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)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{(14 c) \int \frac{1}{(b d+2 c d x)^{7/2} \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac{(42 c) \int \frac{1}{(b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{5 \left (b^2-4 a c\right )^2 d^2}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x}}+\frac{(42 c) \int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx}{5 \left (b^2-4 a c\right )^3 d^4}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x}}+\frac{\left (42 c \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 \left (b^2-4 a c\right )^3 d^4 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x}}+\frac{\left (84 \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 \left (b^2-4 a c\right )^3 d^5 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left (84 \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 \left (b^2-4 a c\right )^{5/2} d^4 \sqrt{a+b x+c x^2}}+\frac{\left (84 \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 \left (b^2-4 a c\right )^{5/2} d^4 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x}}-\frac{84 \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 \left (b^2-4 a c\right )^{9/4} d^{7/2} \sqrt{a+b x+c x^2}}+\frac{\left (84 \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 \left (b^2-4 a c\right )^{5/2} d^4 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}}-\frac{56 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2}}-\frac{168 c \sqrt{a+b x+c x^2}}{5 \left (b^2-4 a c\right )^3 d^3 \sqrt{b d+2 c d x}}+\frac{84 \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 \left (b^2-4 a c\right )^{9/4} d^{7/2} \sqrt{a+b x+c x^2}}-\frac{84 \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 \left (b^2-4 a c\right )^{9/4} d^{7/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0699083, size = 98, normalized size = 0.3 \[ \frac{8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (-\frac{5}{4},\frac{3}{2};-\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{5 d \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.237, size = 876, 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{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 (\frac{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{16 \, c^{6} d^{4} x^{8} + 64 \, b c^{5} d^{4} x^{7} + 8 \,{\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{6} + a^{2} b^{4} d^{4} + 8 \,{\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{5} +{\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{4} + 2 \,{\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{3} +{\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{2} + 2 \,{\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x}, 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 \frac{1}{{\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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