Optimal. Leaf size=429 \[ -\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
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Rubi [A] time = 1.35614, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {911, 96, 93, 208} \[ -\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \sqrt{d+e x} (e f-d g) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
Antiderivative was successfully verified.
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Rule 911
Rule 96
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt{f+g x} \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{2 c}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt{f+g x}}-\frac{2 c}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt{f+g x}}\right ) \, dx\\ &=\frac{(2 c) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt{f+g x}} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt{f+g x}} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt{d+e x}}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt{d+e x}}+\frac{\left (4 c^2\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{\sqrt{b^2-4 a c} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right )}-\frac{\left (4 c^2\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{\sqrt{b^2-4 a c} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right )}\\ &=\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt{d+e x}}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt{d+e x}}+\frac{\left (8 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e-\left (-2 c f+\left (b-\sqrt{b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right )}-\frac{\left (8 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e-\left (-2 c f+\left (b+\sqrt{b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right )}\\ &=\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt{d+e x}}-\frac{4 c e \sqrt{f+g x}}{\sqrt{b^2-4 a c} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt{d+e x}}-\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{f+g x}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right )^{3/2} \sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}+\frac{8 c^2 \tanh ^{-1}\left (\frac{\sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \sqrt{f+g x}}\right )}{\sqrt{b^2-4 a c} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right )^{3/2} \sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}}\\ \end{align*}
Mathematica [A] time = 2.11857, size = 340, normalized size = 0.79 \[ \frac{4 c \left (\frac{e^2 \sqrt{b^2-4 a c} \sqrt{f+g x}}{2 c \sqrt{d+e x} (d g-e f) \left (e (a e-b d)+c d^2\right )}-\frac{2 c \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{g \sqrt{b^2-4 a c}-b g+2 c f}}{\sqrt{f+g x} \sqrt{-e \sqrt{b^2-4 a c}+b e-2 c d}}\right )}{\left (e \left (b-\sqrt{b^2-4 a c}\right )-2 c d\right )^{3/2} \sqrt{g \left (\sqrt{b^2-4 a c}-b\right )+2 c f}}+\frac{2 c \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}}\right )}{\left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right )^{3/2} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.916, size = 47351, normalized size = 110.4 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{\frac{3}{2}} \sqrt{g x + f}}\,{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(-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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