Optimal. Leaf size=454 \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x+c x^2}}{a^2 d x \left (b^2-4 a c\right )}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2} d}-\frac{2 f \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a d x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f^2 \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{f^2 \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]
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Rubi [A] time = 1.19335, antiderivative size = 454, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6725, 740, 806, 724, 206, 975, 1033} \[ -\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x+c x^2}}{a^2 d x \left (b^2-4 a c\right )}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2} d}-\frac{2 f \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac{2 \left (-2 a c+b^2+b c x\right )}{a d x \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f^2 \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right )^{3/2}}+\frac{f^2 \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^{3/2} \left (a f+b \sqrt{d} \sqrt{f}+c d\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6725
Rule 740
Rule 806
Rule 724
Rule 206
Rule 975
Rule 1033
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx &=\int \left (\frac{1}{d x^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{f}{d \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{1}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{d}+\frac{f \int \frac{1}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx}{d}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d x \sqrt{a+b x+c x^2}}-\frac{2 f \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) d \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} \left (-3 b^2+8 a c\right )-b c x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{a \left (b^2-4 a c\right ) d}-\frac{(2 f) \int \frac{\frac{1}{2} \left (b^2-4 a c\right ) f (c d+a f)-\frac{1}{2} b \left (b^2-4 a c\right ) f^2 x}{\sqrt{a+b x+c x^2} \left (d-f x^2\right )} \, dx}{\left (b^2-4 a c\right ) d \left (b^2 d f-(c d+a f)^2\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d x \sqrt{a+b x+c x^2}}-\frac{2 f \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) d \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) d x}-\frac{(3 b) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{2 a^2 d}-\frac{f^{5/2} \int \frac{1}{\left (-\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 d^{3/2} \left (c d-b \sqrt{d} \sqrt{f}+a f\right )}+\frac{f^{5/2} \int \frac{1}{\left (\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 d^{3/2} \left (c d+b \sqrt{d} \sqrt{f}+a f\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d x \sqrt{a+b x+c x^2}}-\frac{2 f \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) d \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) d x}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{a^2 d}+\frac{f^{5/2} \operatorname{Subst}\left (\int \frac{1}{4 c d f-4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{b \sqrt{d} \sqrt{f}-2 a f-\left (-2 c \sqrt{d} \sqrt{f}+b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{d^{3/2} \left (c d-b \sqrt{d} \sqrt{f}+a f\right )}-\frac{f^{5/2} \operatorname{Subst}\left (\int \frac{1}{4 c d f+4 b \sqrt{d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac{-b \sqrt{d} \sqrt{f}-2 a f-\left (2 c \sqrt{d} \sqrt{f}+b f\right ) x}{\sqrt{a+b x+c x^2}}\right )}{d^{3/2} \left (c d+b \sqrt{d} \sqrt{f}+a f\right )}\\ &=\frac{2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d x \sqrt{a+b x+c x^2}}-\frac{2 f \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) d \left (b^2 d f-(c d+a f)^2\right ) \sqrt{a+b x+c x^2}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) d x}+\frac{3 b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{5/2} d}+\frac{f^2 \tanh ^{-1}\left (\frac{b \sqrt{d}-2 a \sqrt{f}+\left (2 c \sqrt{d}-b \sqrt{f}\right ) x}{2 \sqrt{c d-b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 d^{3/2} \left (c d-b \sqrt{d} \sqrt{f}+a f\right )^{3/2}}+\frac{f^2 \tanh ^{-1}\left (\frac{b \sqrt{d}+2 a \sqrt{f}+\left (2 c \sqrt{d}+b \sqrt{f}\right ) x}{2 \sqrt{c d+b \sqrt{d} \sqrt{f}+a f} \sqrt{a+b x+c x^2}}\right )}{2 d^{3/2} \left (c d+b \sqrt{d} \sqrt{f}+a f\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.34166, size = 488, normalized size = 1.07 \[ \frac{\frac{2 \left (8 a c-3 b^2\right ) \sqrt{a+x (b+c x)}}{a^2 x}+\frac{3 b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{a^{5/2}}-\frac{4 f \left (-b c (3 a f+c d)-2 c^2 x (a f+c d)+b^2 c f x+b^3 f\right )}{\sqrt{a+x (b+c x)} \left (b^2 d f-(a f+c d)^2\right )}-\frac{f^2 \left (\frac{\left (b^2-4 a c\right ) \left (a f+b \sqrt{d} \sqrt{f}+c d\right ) \tanh ^{-1}\left (\frac{2 a \sqrt{f}-b \sqrt{d}+b \sqrt{f} x-2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{\left (4 a c-b^2\right ) \left (a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d\right ) \tanh ^{-1}\left (\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\sqrt{d} \left ((a f+c d)^2-b^2 d f\right )}+\frac{4 \left (-2 a c+b^2+b c x\right )}{a x \sqrt{a+x (b+c x)}}}{2 d \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.25, size = 1656, normalized size = 3.7 \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{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (f x^{2} - d\right )} x^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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