Optimal. Leaf size=614 \[ -\frac {a^{3/2} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {b f \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c d^2}-\frac {b \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}+\frac {f \left (8 a c+b^2+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {3 b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d} \]
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Rubi [A] time = 1.44, antiderivative size = 614, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {6725, 732, 812, 843, 621, 206, 724, 734, 814, 1021, 1070, 1078, 1033} \[ -\frac {a^{3/2} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c d^2}-\frac {b f \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (12 a c f+b^2 (-f)+24 c^2 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}+\frac {f \left (8 a c+b^2+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {3 b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 732
Rule 734
Rule 812
Rule 814
Rule 843
Rule 1021
Rule 1033
Rule 1070
Rule 1078
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx &=\int \left (\frac {\left (a+b x+c x^2\right )^{3/2}}{d x^3}+\frac {f \left (a+b x+c x^2\right )^{3/2}}{d^2 x}+\frac {f^2 x \left (a+b x+c x^2\right )^{3/2}}{d^2 \left (d-f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx}{d}+\frac {f \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x} \, dx}{d^2}+\frac {f^2 \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx}{d^2}\\ &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}+\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x^2} \, dx}{4 d}+\frac {f \int \frac {\sqrt {a+b x+c x^2} \left (\frac {3 b d}{2}+3 (c d+a f) x+\frac {3}{2} b f x^2\right )}{d-f x^2} \, dx}{3 d^2}-\frac {f \int \frac {(-2 a-b x) \sqrt {a+b x+c x^2}}{x} \, dx}{2 d^2}\\ &=-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \int \frac {-b^2-4 a c-4 b c x}{x \sqrt {a+b x+c x^2}} \, dx}{8 d}-\frac {\int \frac {-\frac {3}{8} b d f \left (8 c^2 d+b^2 f+20 a c f\right )-6 c f \left (b^2 d f+(c d+a f)^2\right ) x-\frac {3}{8} b f^2 \left (24 c^2 d-b^2 f+12 a c f\right ) x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{6 c d^2 f}+\frac {f \int \frac {8 a^2 c-\frac {1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{8 c d^2}\\ &=-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}+\frac {(3 b c) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 d}+\frac {\left (3 \left (b^2+4 a c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{8 d}+\frac {\int \frac {\frac {3}{8} b d f^2 \left (24 c^2 d-b^2 f+12 a c f\right )+\frac {3}{8} b d f^2 \left (8 c^2 d+b^2 f+20 a c f\right )+6 c f^2 \left (b^2 d f+(c d+a f)^2\right ) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{6 c d^2 f^2}+\frac {\left (a^2 f\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{d^2}-\frac {\left (b \left (b^2-12 a c\right ) f\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c d^2}-\frac {\left (b \left (24 c^2 d-b^2 f+12 a c f\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c d^2}\\ &=-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}+\frac {(3 b c) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{d}-\frac {\left (3 \left (b^2+4 a c\right )\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 d}-\frac {\left (2 a^2 f\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 )}{d^2}-\frac {\left (b \left (b^2-12 a c\right ) f\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c d^2}+\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^2 \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d^2}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^2 \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 d^2}-\frac {\left (b \left (24 c^2 d-b^2 f+12 a c f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c d^2}\\ &=-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \left (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 \sqrt {a} d}-\frac {a^{3/2} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {b \left (b^2-12 a c\right ) f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^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 {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^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 {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \left (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 \sqrt {a} d}-\frac {a^{3/2} f \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 b \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {b \left (b^2-12 a c\right ) f \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{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 {f}}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{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 {f}}\\ \end {align*}
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Mathematica [A] time = 0.97, size = 303, normalized size = 0.49 \[ -\frac {\frac {\left (4 a (2 a f+3 c d)+3 b^2 d\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{\sqrt {a}}-\frac {4 \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {2 a \sqrt {f}-b \sqrt {d}+b \sqrt {f} x-2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{\sqrt {f}}+\frac {4 \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \tanh ^{-1}\left (\frac {-2 \left (a \sqrt {f}+c \sqrt {d} x\right )-b \left (\sqrt {d}+\sqrt {f} x\right )}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{\sqrt {f}}+\frac {2 d (2 a+5 b x) \sqrt {a+x (b+c x)}}{x^2}}{8 d^2} \]
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 = 5056, normalized size = 8.23 \[ \text {output too large to display} \]
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 \[ -\int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} - d\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^3\,\left (d-f\,x^2\right )} \,d x \]
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 \[ - \int \frac {a \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {b x \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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