Optimal. Leaf size=176 \[ \frac{80 c^2 \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}+\frac{40 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{7/2}}+\frac{40 c}{3 d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.105792, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {687, 693, 688, 205} \[ \frac{80 c^2 \sqrt{a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}+\frac{40 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{7/2}}+\frac{40 c}{3 d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \sqrt{a+b x+c x^2}}-\frac{2}{3 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{(20 c) \int \frac{1}{(b d+2 c d x)^3 \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^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}+\frac{\left (80 c^2\right ) \int \frac{1}{(b d+2 c d x)^3 \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^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}+\frac{\left (40 c^2\right ) \int \frac{1}{(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^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}+\frac{\left (160 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )}{\left (b^2-4 a c\right )^3 d^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}+\frac{40 c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} d^3}\\ \end{align*}
Mathematica [C] time = 0.0364234, size = 62, normalized size = 0.35 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{3 d^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 267, normalized size = 1.5 \begin{align*} -{\frac{1}{4\,{c}^{2}{d}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( x+{\frac{b}{2\,c}} \right ) ^{-2} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{-{\frac{3}{2}}}}-{\frac{5}{3\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{-{\frac{3}{2}}}}-20\,{\frac{c}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}}{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}+40\,{\frac{c}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}}\ln \left ({ \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 40.8697, size = 2759, normalized size = 15.68 \begin{align*} \text{result too large to display} \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*} \frac{\int \frac{1}{a^{2} b^{3} \sqrt{a + b x + c x^{2}} + 6 a^{2} b^{2} c x \sqrt{a + b x + c x^{2}} + 12 a^{2} b c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 8 a^{2} c^{3} x^{3} \sqrt{a + b x + c x^{2}} + 2 a b^{4} x \sqrt{a + b x + c x^{2}} + 14 a b^{3} c x^{2} \sqrt{a + b x + c x^{2}} + 36 a b^{2} c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 40 a b c^{3} x^{4} \sqrt{a + b x + c x^{2}} + 16 a c^{4} x^{5} \sqrt{a + b x + c x^{2}} + b^{5} x^{2} \sqrt{a + b x + c x^{2}} + 8 b^{4} c x^{3} \sqrt{a + b x + c x^{2}} + 25 b^{3} c^{2} x^{4} \sqrt{a + b x + c x^{2}} + 38 b^{2} c^{3} x^{5} \sqrt{a + b x + c x^{2}} + 28 b c^{4} x^{6} \sqrt{a + b x + c x^{2}} + 8 c^{5} x^{7} \sqrt{a + b x + c x^{2}}}\, dx}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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