Optimal. Leaf size=376 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{5/2} d}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}-\frac{f^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{f^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{f \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2} \]
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Rubi [A] time = 0.734105, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6725, 744, 806, 724, 206, 1033} \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{5/2} d}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}-\frac{f^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+x \left (2 c \sqrt{d}-b \sqrt{f}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}+\frac{f^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+x \left (b \sqrt{f}+2 c \sqrt{d}\right )+b \sqrt{d}}{2 \sqrt{a+b x+c x^2} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{2 d^2 \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{f \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2} \]
Antiderivative was successfully verified.
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Rule 6725
Rule 744
Rule 806
Rule 724
Rule 206
Rule 1033
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{a+b x+c x^2} \left (d-f x^2\right )} \, dx &=\int \left (\frac{1}{d x^3 \sqrt{a+b x+c x^2}}+\frac{f}{d^2 x \sqrt{a+b x+c x^2}}+\frac{f^2 x}{d^2 \sqrt{a+b x+c x^2} \left (d-f x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{1}{x^3 \sqrt{a+b x+c x^2}} \, dx}{d}+\frac{f \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{d^2}+\frac{f^2 \int \frac{x}{\sqrt{a+b x+c x^2} \left (d-f x^2\right )} \, dx}{d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}-\frac{\int \frac{\frac{3 b}{2}+c x}{x^2 \sqrt{a+b x+c x^2}} \, dx}{2 a d}-\frac{(2 f) \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 )}{d^2}+\frac{f^2 \int \frac{1}{\left (-\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 d^2}+\frac{f^2 \int \frac{1}{\left (\sqrt{d} \sqrt{f}-f x\right ) \sqrt{a+b x+c x^2}} \, dx}{2 d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}-\frac{f \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d^2}+\frac{\left (3 b^2-4 a c\right ) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{8 a^2 d}-\frac{f^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^2}-\frac{f^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^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}-\frac{f \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d^2}-\frac{f^{3/2} \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^2 \sqrt{c d-b \sqrt{d} \sqrt{f}+a f}}+\frac{f^{3/2} \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^2 \sqrt{c d+b \sqrt{d} \sqrt{f}+a f}}-\frac{\left (3 b^2-4 a c\right ) \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 )}{4 a^2 d}\\ &=-\frac{\sqrt{a+b x+c x^2}}{2 a d x^2}+\frac{3 b \sqrt{a+b x+c x^2}}{4 a^2 d x}-\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{5/2} d}-\frac{f \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} d^2}-\frac{f^{3/2} \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^2 \sqrt{c d-b \sqrt{d} \sqrt{f}+a f}}+\frac{f^{3/2} \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^2 \sqrt{c d+b \sqrt{d} \sqrt{f}+a f}}\\ \end{align*}
Mathematica [A] time = 2.15525, size = 314, normalized size = 0.84 \[ \frac{2 \sqrt{a} \left (\frac{2 a^2 f^{3/2} \tanh ^{-1}\left (\frac{2 a \sqrt{f}+b \sqrt{d}+b \sqrt{f} x+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}\right )}{\sqrt{a f+b \sqrt{d} \sqrt{f}+c d}}-\frac{2 a^2 f^{3/2} \tanh ^{-1}\left (\frac{-2 a \sqrt{f}+b \left (\sqrt{d}-\sqrt{f} x\right )+2 c \sqrt{d} x}{2 \sqrt{a+x (b+c x)} \sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}\right )}{\sqrt{a f+b \left (-\sqrt{d}\right ) \sqrt{f}+c d}}-\frac{d (2 a-3 b x) \sqrt{a+x (b+c x)}}{x^2}\right )+\left (4 a (c d-2 a f)-3 b^2 d\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{8 a^{5/2} d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.279, size = 519, normalized size = 1.4 \begin{align*} -{\frac{f}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{f}{2\,{d}^{2}}\ln \left ({ \left ( 2\,{\frac{-b\sqrt{df}+af+cd}{f}}+{\frac{1}{f} \left ( -2\,c\sqrt{df}+bf \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{-b\sqrt{df}+af+cd}{f}}}\sqrt{ \left ( x+{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{-2\,c\sqrt{df}+bf}{f} \left ( x+{\frac{\sqrt{df}}{f}} \right ) }+{\frac{-b\sqrt{df}+af+cd}{f}}} \right ) \left ( x+{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{1}{f} \left ( -b\sqrt{df}+af+cd \right ) }}}}}-{\frac{1}{2\,ad{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,b}{4\,{a}^{2}dx}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}}{8\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{c}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}+{\frac{f}{2\,{d}^{2}}\ln \left ({ \left ( 2\,{\frac{b\sqrt{df}+af+cd}{f}}+{\frac{1}{f} \left ( 2\,c\sqrt{df}+bf \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) }+2\,\sqrt{{\frac{b\sqrt{df}+af+cd}{f}}}\sqrt{ \left ( x-{\frac{\sqrt{df}}{f}} \right ) ^{2}c+{\frac{2\,c\sqrt{df}+bf}{f} \left ( x-{\frac{\sqrt{df}}{f}} \right ) }+{\frac{b\sqrt{df}+af+cd}{f}}} \right ) \left ( x-{\frac{1}{f}\sqrt{df}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{1}{f} \left ( b\sqrt{df}+af+cd \right ) }}}}} \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}{\sqrt{c x^{2} + b x + a}{\left (f x^{2} - d\right )} x^{3}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- d x^{3} \sqrt{a + b x + c x^{2}} + f x^{5} \sqrt{a + b x + c x^{2}}}\, dx \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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