Optimal. Leaf size=307 \[ -\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{8 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^4 \left (a e^2-b d e+c d^2\right )^{3/2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.355388, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {732, 810, 843, 621, 206, 724} \[ -\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{8 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^4 \left (a e^2-b d e+c d^2\right )^{3/2}}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 732
Rule 810
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx &=-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{2 e}\\ &=-\frac{\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}-\frac{\int \frac{\frac{1}{2} \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )-8 c^2 \left (c d^2-b d e+a e^2\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{8 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{c^2 \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{e^4}-\frac{\left (8 c^2 d \left (c d^2-b d e+a e^2\right )+\frac{1}{2} e \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{8 e^4 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{\left (2 c^2\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 )}{e^4}+\frac{\left (8 c^2 d \left (c d^2-b d e+a e^2\right )+\frac{1}{2} e \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{4 e^4 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^4}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{16 e^4 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.71053, size = 421, normalized size = 1.37 \[ \frac{\frac{3 \left (\frac{2 (a+x (b+c x))^{3/2} \left (4 c e (b d-2 a e)+b^2 e^2-4 c^2 d^2\right )}{d+e x}+\frac{2 \sqrt{a+x (b+c x)} \left (2 c^2 e (2 a e (2 e x-3 d)+b d (7 d-2 e x))-b c e^2 (-10 a e+5 b d+b e x)-b^3 e^3+4 c^3 d^2 (e x-2 d)\right )}{e^2}+\frac{(2 c d-b e) \left (4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right ) \sqrt{e (a e-b d)+c d^2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )+16 c^{3/2} \left (e (a e-b d)+c d^2\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{e^3}\right )}{8 \left (e (a e-b d)+c d^2\right )^2}+\frac{3 (a+x (b+c x))^{3/2} (2 c d-b e)}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}-\frac{2 (a+x (b+c x))^{3/2}}{(d+e x)^3}}{6 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.246, size = 10401, normalized size = 33.9 \begin{align*} \text{output too large to display} \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 [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(-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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