Optimal. Leaf size=392 \[ -\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} f}+\frac {b (b+2 c x) \sqrt {a+b x+c x^2}}{8 c^2 f}-\frac {d \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^{5/2}}+\frac {d \sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^{5/2}}-\frac {d \sqrt {a+b x+c x^2}}{f^2}-\frac {b d \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c f} \]
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Rubi [A] time = 0.95, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {6725, 640, 612, 621, 206, 1021, 1078, 1033, 724} \[ -\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} f}+\frac {b (b+2 c x) \sqrt {a+b x+c x^2}}{8 c^2 f}-\frac {d \sqrt {a+b x+c x^2}}{f^2}-\frac {b d \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f^2}-\frac {d \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^{5/2}}+\frac {d \sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^{5/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 724
Rule 1021
Rule 1033
Rule 1078
Rule 6725
Rubi steps
\begin {align*} \int \frac {x^3 \sqrt {a+b x+c x^2}}{d-f x^2} \, dx &=\int \left (-\frac {x \sqrt {a+b x+c x^2}}{f}+\frac {d x \sqrt {a+b x+c x^2}}{f \left (d-f x^2\right )}\right ) \, dx\\ &=-\frac {\int x \sqrt {a+b x+c x^2} \, dx}{f}+\frac {d \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx}{f}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{f^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c f}+\frac {d \int \frac {\frac {b d}{2}+(c d+a f) x+\frac {1}{2} b f x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{f^2}+\frac {b \int \sqrt {a+b x+c x^2} \, dx}{2 c f}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{f^2}+\frac {b (b+2 c x) \sqrt {a+b x+c x^2}}{8 c^2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c f}-\frac {d \int \frac {-b d f-f (c d+a f) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{f^3}-\frac {(b d) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 f^2}-\frac {\left (b \left (b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^2 f}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{f^2}+\frac {b (b+2 c x) \sqrt {a+b x+c x^2}}{8 c^2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c f}-\frac {(b d) \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 )}{f^2}-\frac {\left (b \left (b^2-4 a c\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^2 f}+\frac {\left (d \left (c d-b \sqrt {d} \sqrt {f}+a f\right )\right ) \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f^2}+\frac {\left (d \left (c d+b \sqrt {d} \sqrt {f}+a f\right )\right ) \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f^2}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{f^2}+\frac {b (b+2 c x) \sqrt {a+b x+c x^2}}{8 c^2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c f}-\frac {b d \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f^2}-\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} f}-\frac {\left (d \left (c d-b \sqrt {d} \sqrt {f}+a f\right )\right ) \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 )}{f^2}-\frac {\left (d \left (c d+b \sqrt {d} \sqrt {f}+a f\right )\right ) \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 )}{f^2}\\ &=-\frac {d \sqrt {a+b x+c x^2}}{f^2}+\frac {b (b+2 c x) \sqrt {a+b x+c x^2}}{8 c^2 f}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c f}-\frac {b d \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f^2}-\frac {b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} f}-\frac {d \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \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 f^{5/2}}+\frac {d \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \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 f^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 327, normalized size = 0.83 \[ \frac {-\frac {2 \sqrt {f} \sqrt {a+x (b+c x)} \left (2 c f (4 a+b x)-3 b^2 f+8 c^2 \left (3 d+f x^2\right )\right )}{c^2}-\frac {3 b \sqrt {f} \left (-4 a c f+b^2 f+8 c^2 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{5/2}}+24 d \sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+b \sqrt {d}+b \sqrt {f} x+2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )-24 d \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+b \left (\sqrt {d}-\sqrt {f} x\right )+2 c \sqrt {d} x}{2 \sqrt {a+x (b+c x)} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{48 f^{5/2}} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1817, normalized size = 4.64 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^3\,\sqrt {c\,x^2+b\,x+a}}{d-f\,x^2} \,d x \]
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 \[ - \int \frac {x^{3} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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