Optimal. Leaf size=394 \[ -\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2} d}-\frac {2 f \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {f^{3/2} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {f^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}} \]
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Rubi [A] time = 1.18, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6725, 740, 12, 724, 206, 1018, 1033} \[ -\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2} d}-\frac {2 f \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {f^{3/2} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {f^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 740
Rule 1018
Rule 1033
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx &=\int \left (\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}-\frac {f x}{d \left (a+b x+c x^2\right )^{3/2} \left (-d+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{x \left (a+b x+c x^2\right )^{3/2}} \, dx}{d}-\frac {f \int \frac {x}{\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 \sqrt {a+b x+c x^2}}-\frac {2 f \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) 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 {b^2}{2}+2 a c}{x \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} b \left (b^2-4 a c\right ) d f-\frac {1}{2} \left (b^2-4 a c\right ) f (c d+a f) 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 \sqrt {a+b x+c x^2}}-\frac {2 f \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) 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 {\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{a d}-\frac {f^2 \int \frac {1}{\left (\sqrt {d} \sqrt {f}+f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d \left (c d-b \sqrt {d} \sqrt {f}+a f\right )}-\frac {f^2 \int \frac {1}{\left (-\sqrt {d} \sqrt {f}+f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d \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 \sqrt {a+b x+c x^2}}-\frac {2 f \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) 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 \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 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 \left (c d-b \sqrt {d} \sqrt {f}+a f\right )}+\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 \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 \sqrt {a+b x+c x^2}}-\frac {2 f \left (a \left (2 c^2 d-b^2 f+2 a c f\right )+b c (c d-a f) 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 {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2} d}-\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 \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\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 \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 436, normalized size = 1.11 \[ \frac {-\frac {\tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{a^{3/2}}-\frac {2 f \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \left (b^2 d f-(a f+c d)^2\right )}+\frac {f^{3/2} \left (\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+b \sqrt {d}+b \sqrt {f} x+2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )+\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}-b \sqrt {d}+b \sqrt {f} x-2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )\right )}{2 \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \left ((a f+c d)^2-b^2 d f\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}}{d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1518, normalized size = 3.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (f x^{2} - d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x\,\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{- a d x \sqrt {a + b x + c x^{2}} + a f x^{3} \sqrt {a + b x + c x^{2}} - b d x^{2} \sqrt {a + b x + c x^{2}} + b f x^{4} \sqrt {a + b x + c x^{2}} - c d x^{3} \sqrt {a + b x + c x^{2}} + c f x^{5} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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