Optimal. Leaf size=607 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-2 a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left (-a b \left (e \left (2 d \sqrt{b^2-4 a c}-a e\right )+3 c d^2\right )+a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )\right )+b^2 d \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \sqrt{c} \left (-a b \left (3 c d^2-e \left (2 d \sqrt{b^2-4 a c}+a e\right )\right )-a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )\right )-b^2 d \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x} (b d-2 a e)}{a^2 x}-\frac{e (b d-2 a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^2 \sqrt{d}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 a \sqrt{d}}-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a x} \]
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Rubi [A] time = 3.93064, antiderivative size = 607, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {897, 1287, 199, 206, 1166, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-2 a b d e-a \left (c d^2-a e^2\right )+b^2 d^2\right )}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left (-a b \left (e \left (2 d \sqrt{b^2-4 a c}-a e\right )+3 c d^2\right )+a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )\right )+b^2 d \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{2} \sqrt{c} \left (-a b \left (3 c d^2-e \left (2 d \sqrt{b^2-4 a c}+a e\right )\right )-a \left (a e^2 \sqrt{b^2-4 a c}-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )\right )-b^2 d \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x} (b d-2 a e)}{a^2 x}-\frac{e (b d-2 a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^2 \sqrt{d}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 a \sqrt{d}}-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a x} \]
Antiderivative was successfully verified.
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Rule 897
Rule 1287
Rule 199
Rule 206
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{x^3 \left (a+b x+c x^2\right )} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^4}{\left (-\frac{d}{e}+\frac{x^2}{e}\right )^3 \left (\frac{c d^2-b d e+a e^2}{e^2}-\frac{(2 c d-b e) x^2}{e^2}+\frac{c x^4}{e^2}\right )} \, dx,x,\sqrt{d+e x}\right )}{e}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{d^2 e^3}{a \left (d-x^2\right )^3}+\frac{d e^2 (-b d+2 a e)}{a^2 \left (d-x^2\right )^2}+\frac{e \left (-b^2 d^2+2 a b d e+a \left (c d^2-a e^2\right )\right )}{a^3 \left (d-x^2\right )}+\frac{e \left (\left (b^2 d-a c d-a b e\right ) \left (c d^2-b d e+a e^2\right )-c \left (b^2 d^2-2 a b d e-a \left (c d^2-a e^2\right )\right ) x^2\right )}{a^3 \left (c d^2-b d e+a e^2-(2 c d-b e) x^2+c x^4\right )}\right ) \, dx,x,\sqrt{d+e x}\right )}{e}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (b^2 d-a c d-a b e\right ) \left (c d^2-b d e+a e^2\right )-c \left (b^2 d^2-2 a b d e-a \left (c d^2-a e^2\right )\right ) 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 )}{a^3}-\frac{\left (2 d^2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d-x^2\right )^3} \, dx,x,\sqrt{d+e x}\right )}{a}-\frac{(2 d e (b d-2 a e)) \operatorname{Subst}\left (\int \frac{1}{\left (d-x^2\right )^2} \, dx,x,\sqrt{d+e x}\right )}{a^2}-\frac{\left (2 \left (b^2 d^2-2 a b d e-a \left (c d^2-a e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{d-x^2} \, dx,x,\sqrt{d+e x}\right )}{a^3}\\ &=-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{(b d-2 a e) \sqrt{d+e x}}{a^2 x}-\frac{2 \left (b^2 d^2-2 a b d e-a \left (c d^2-a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^3 \sqrt{d}}-\frac{\left (3 d e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d-x^2\right )^2} \, dx,x,\sqrt{d+e x}\right )}{2 a}-\frac{(e (b d-2 a e)) \operatorname{Subst}\left (\int \frac{1}{d-x^2} \, dx,x,\sqrt{d+e x}\right )}{a^2}-\frac{\left (c \left (b^3 d^2+b^2 d \left (\sqrt{b^2-4 a c} d-2 a e\right )+a \left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-4 a e\right )\right )-a b \left (3 c d^2+e \left (2 \sqrt{b^2-4 a c} d-a e\right )\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 )}{a^3 \sqrt{b^2-4 a c}}+\frac{\left (c \left (b^3 d^2-b^2 d \left (\sqrt{b^2-4 a c} d+2 a e\right )-a b \left (3 c d^2-e \left (2 \sqrt{b^2-4 a c} d+a e\right )\right )-a \left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+4 a e\right )\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 )}{a^3 \sqrt{b^2-4 a c}}\\ &=-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a x}+\frac{(b d-2 a e) \sqrt{d+e x}}{a^2 x}-\frac{e (b d-2 a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^2 \sqrt{d}}-\frac{2 \left (b^2 d^2-2 a b d e-a \left (c d^2-a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left (b^3 d^2+b^2 d \left (\sqrt{b^2-4 a c} d-2 a e\right )+a \left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-4 a e\right )\right )-a b \left (3 c d^2+e \left (2 \sqrt{b^2-4 a c} d-a e\right )\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 )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{\sqrt{2} \sqrt{c} \left (b^3 d^2-b^2 d \left (\sqrt{b^2-4 a c} d+2 a e\right )-a b \left (3 c d^2-e \left (2 \sqrt{b^2-4 a c} d+a e\right )\right )-a \left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+4 a e\right )\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 )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}-\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{d-x^2} \, dx,x,\sqrt{d+e x}\right )}{4 a}\\ &=-\frac{d \sqrt{d+e x}}{2 a x^2}+\frac{3 e \sqrt{d+e x}}{4 a x}+\frac{(b d-2 a e) \sqrt{d+e x}}{a^2 x}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 a \sqrt{d}}-\frac{e (b d-2 a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^2 \sqrt{d}}-\frac{2 \left (b^2 d^2-2 a b d e-a \left (c d^2-a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{a^3 \sqrt{d}}+\frac{\sqrt{2} \sqrt{c} \left (b^3 d^2+b^2 d \left (\sqrt{b^2-4 a c} d-2 a e\right )+a \left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-4 a e\right )\right )-a b \left (3 c d^2+e \left (2 \sqrt{b^2-4 a c} d-a e\right )\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 )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{\sqrt{2} \sqrt{c} \left (b^3 d^2-b^2 d \left (\sqrt{b^2-4 a c} d+2 a e\right )-a b \left (3 c d^2-e \left (2 \sqrt{b^2-4 a c} d+a e\right )\right )-a \left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+4 a e\right )\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 )}{a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\\ \end{align*}
Mathematica [A] time = 2.87339, size = 587, normalized size = 0.97 \[ \frac{-\frac{2 a^2 d \sqrt{d+e x}}{x^2}+3 a^2 e \left (\frac{\sqrt{d+e x}}{x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}\right )-\frac{8 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-2 a b d e+a \left (a e^2-c d^2\right )+b^2 d^2\right )}{\sqrt{d}}+\frac{4 \sqrt{2} \sqrt{c} \left (a b \left (e \left (a e-2 d \sqrt{b^2-4 a c}\right )-3 c d^2\right )+a \left (c d \left (4 a e-d \sqrt{b^2-4 a c}\right )+a e^2 \sqrt{b^2-4 a c}\right )+b^2 d \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{4 \sqrt{2} \sqrt{c} \left (a b \left (e \left (2 d \sqrt{b^2-4 a c}+a e\right )-3 c d^2\right )+a \left (c d \left (d \sqrt{b^2-4 a c}+4 a e\right )-a e^2 \sqrt{b^2-4 a c}\right )-b^2 d \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{4 a \sqrt{d+e x} (b d-2 a e)}{x}+\frac{4 a e (2 a e-b d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.304, size = 1880, normalized size = 3.1 \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(-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(-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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