Optimal. Leaf size=469 \[ \frac{3 \sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{3 \sqrt{c} e \left (-2 c e \left (-d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{3 e^2 (2 c d-b e)}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2} \]
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Rubi [A] time = 1.7209, antiderivative size = 469, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {822, 828, 826, 1166, 208} \[ \frac{3 \sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{3 \sqrt{c} e \left (-2 c e \left (-d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{3 e^2 (2 c d-b e)}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 822
Rule 828
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx &=-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \left (a+b x+c x^2\right )}-\frac{\int \frac{\frac{3}{2} \left (b^2-4 a c\right ) e (c d-b e)-\frac{3}{2} c \left (b^2-4 a c\right ) e^2 x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=\frac{3 e^2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \left (a+b x+c x^2\right )}-\frac{\int \frac{\frac{3}{2} \left (b^2-4 a c\right ) e \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-\frac{3}{2} c \left (b^2-4 a c\right ) e^2 (2 c d-b e) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{3 e^2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \left (a+b x+c x^2\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{3}{2} c \left (b^2-4 a c\right ) d e^2 (2 c d-b e)+\frac{3}{2} \left (b^2-4 a c\right ) e^2 \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-\frac{3}{2} c \left (b^2-4 a c\right ) e^2 (2 c d-b e) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{3 e^2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \left (a+b x+c x^2\right )}+\frac{\left (3 c e \left (2 c^2 d^2+b \left (b-\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt{b^2-4 a c} d+a e\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}-\frac{\left (3 c e \left (2 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt{b^2-4 a c} d+a e\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{3 e^2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \left (a+b x+c x^2\right )}+\frac{3 \sqrt{c} e \left (2 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt{b^2-4 a c} d+a e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}-\frac{3 \sqrt{c} e \left (2 c^2 d^2+b \left (b-\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt{b^2-4 a c} d+a e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 3.17464, size = 381, normalized size = 0.81 \[ \frac{\frac{3 \sqrt{2} \sqrt{c} e \left (\frac{\left (-2 c e \left (d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{\left (-2 c e \left (-d \sqrt{b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c}}+\frac{2 \left (e (a e-b d)+c d^2\right ) (b e-c d+c e x)}{\sqrt{d+e x} (a+x (b+c x))}-\frac{6 e^2 (b e-2 c d)}{\sqrt{d+e x}}}{2 \left (e (a e-b d)+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 1758, normalized size = 3.8 \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{2 \, c x + b}{{\left (c x^{2} + b x + a\right )}^{2}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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(-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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