Optimal. Leaf size=177 \[ \frac{e \sqrt{a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2}} \]
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Rubi [A] time = 0.177996, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {738, 779, 621, 206} \[ \frac{e \sqrt{a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}-\frac{2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 738
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{(d+e x) (-2 e (b d-2 a e)-2 e (2 c d-b e) x)}{\sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{e \left (8 c^2 d^2+3 b^2 e^2-2 c e (3 b d+4 a e)+2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}+\frac{\left (3 e^2 (2 c d-b e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 c^2}\\ &=-\frac{2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{e \left (8 c^2 d^2+3 b^2 e^2-2 c e (3 b d+4 a e)+2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}+\frac{\left (3 e^2 (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{c^2}\\ &=-\frac{2 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{e \left (8 c^2 d^2+3 b^2 e^2-2 c e (3 b d+4 a e)+2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.418983, size = 196, normalized size = 1.11 \[ \frac{\frac{2 \sqrt{c} \left (4 c \left (2 a^2 e^3+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )+c^2 d^3 x\right )-b^2 e^2 (3 a e+c x (e x-6 d))+2 b c \left (a e^2 (3 d+5 e x)+c d^2 (d-3 e x)\right )-3 b^3 e^3 x\right )}{\sqrt{a+x (b+c x)}}+3 e^2 \left (b^2-4 a c\right ) (b e-2 c d) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{2 c^{5/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 541, normalized size = 3.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,b{e}^{3}x}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{e}^{3}{b}^{2}}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{b}^{3}{e}^{3}x}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{e}^{3}{b}^{4}}{4\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,b{e}^{3}}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{c{x}^{2}+bx+a}}}+4\,{\frac{ab{e}^{3}x}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{a{e}^{3}{b}^{2}}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-3\,{\frac{d{e}^{2}x}{c\sqrt{c{x}^{2}+bx+a}}}+{\frac{3\,d{e}^{2}b}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+3\,{\frac{d{e}^{2}{b}^{2}x}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{3\,{b}^{3}d{e}^{2}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+3\,{\frac{d{e}^{2}}{{c}^{3/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx+a} \right ) }-3\,{\frac{{d}^{2}e}{c\sqrt{c{x}^{2}+bx+a}}}-6\,{\frac{{d}^{2}ebx}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-3\,{\frac{{d}^{2}e{b}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{{d}^{3} \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \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 = 4.84532, size = 1574, normalized size = 8.89 \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*} \int \frac{\left (d + e x\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14462, size = 315, normalized size = 1.78 \begin{align*} \frac{{\left (\frac{{\left (b^{2} c e^{3} - 4 \, a c^{2} e^{3}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} - \frac{4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 12 \, a c^{2} d e^{2} - 3 \, b^{3} e^{3} + 10 \, a b c e^{3}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac{2 \, b c^{2} d^{3} - 12 \, a c^{2} d^{2} e + 6 \, a b c d e^{2} - 3 \, a b^{2} e^{3} + 8 \, a^{2} c e^{3}}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt{c x^{2} + b x + a}} - \frac{3 \,{\left (2 \, c d e^{2} - b e^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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