Optimal. Leaf size=496 \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d}-\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d f^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (\sqrt {e^2-4 d f}+e\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d f^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {c^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{f^2}+\frac {\sqrt {a+c x^2} (c d-a f)}{d f}+\frac {a \sqrt {a+c x^2}}{d} \]
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Rubi [A] time = 2.57, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6728, 266, 50, 63, 208, 1020, 1080, 217, 206, 1034, 725} \[ -\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d f^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (\sqrt {e^2-4 d f}+e\right ) \left (c^2 d e^2-f (c d-a f)^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d f^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d}-\frac {c^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{f^2}+\frac {\sqrt {a+c x^2} (c d-a f)}{d f}+\frac {a \sqrt {a+c x^2}}{d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 725
Rule 1020
Rule 1034
Rule 1080
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2}}{x \left (d+e x+f x^2\right )} \, dx &=\int \left (\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {(-e-f x) \left (a+c x^2\right )^{3/2}}{d \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a+c x^2\right )^{3/2}}{x} \, dx}{d}+\frac {\int \frac {(-e-f x) \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx}{d}\\ &=-\frac {\left (a+c x^2\right )^{3/2}}{3 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x} \, dx,x,x^2\right )}{2 d}+\frac {\int \frac {(-3 a e f+3 f (c d-a f) x) \sqrt {a+c x^2}}{d+e x+f x^2} \, dx}{3 d f}\\ &=\frac {(c d-a f) \sqrt {a+c x^2}}{d f}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d}+\frac {\int \frac {-3 a^2 e f^2-3 f (c d-a f)^2 x-3 c^2 d e f x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{3 d f^2}\\ &=\frac {a \sqrt {a+c x^2}}{d}+\frac {(c d-a f) \sqrt {a+c x^2}}{d f}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d}+\frac {\int \frac {3 c^2 d^2 e f-3 a^2 e f^3+\left (3 c^2 d e^2 f-3 f^2 (c d-a f)^2\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{3 d f^3}-\frac {\left (c^2 e\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{f^2}\\ &=\frac {a \sqrt {a+c x^2}}{d}+\frac {(c d-a f) \sqrt {a+c x^2}}{d f}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d}-\frac {\left (c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{f^2}+\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e-\sqrt {e^2-4 d f}\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d f^2 \sqrt {e^2-4 d f}}-\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e+\sqrt {e^2-4 d f}\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d f^2 \sqrt {e^2-4 d f}}\\ &=\frac {a \sqrt {a+c x^2}}{d}+\frac {(c d-a f) \sqrt {a+c x^2}}{d f}-\frac {c^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{f^2}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d}-\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e-\sqrt {e^2-4 d f}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d f^2 \sqrt {e^2-4 d f}}+\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e+\sqrt {e^2-4 d f}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d f^2 \sqrt {e^2-4 d f}}\\ &=\frac {a \sqrt {a+c x^2}}{d}+\frac {(c d-a f) \sqrt {a+c x^2}}{d f}-\frac {c^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{f^2}-\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e-\sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d f^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (2 e f \left (c^2 d^2-a^2 f^2\right )-\left (c^2 d e^2-f (c d-a f)^2\right ) \left (e+\sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d f^2 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 1.58, size = 746, normalized size = 1.50 \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d}-\frac {c^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{f^2}+\frac {a \sqrt {a f^2+\frac {1}{2} c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {a+c x^2} \sqrt {4 a f^2+2 c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{2 d f}-\frac {a e \sqrt {4 a f^2+2 c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {a+c x^2} \sqrt {4 a f^2+2 c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 d f \sqrt {e^2-4 d f}}-\frac {\left (c d \left (\sqrt {e^2-4 d f}-e\right )-a f \left (\sqrt {e^2-4 d f}+e\right )\right ) \sqrt {4 a f^2-2 c \left (e \sqrt {e^2-4 d f}+2 d f-e^2\right )} \tanh ^{-1}\left (\frac {2 a f+c x \left (\sqrt {e^2-4 d f}-e\right )}{\sqrt {a+c x^2} \sqrt {4 a f^2-2 c \left (e \sqrt {e^2-4 d f}+2 d f-e^2\right )}}\right )}{4 d f^2 \sqrt {e^2-4 d f}}-\frac {c \sqrt {a f^2+\frac {1}{2} c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {a+c x^2} \sqrt {4 a f^2+2 c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{2 f^2}-\frac {c e \sqrt {4 a f^2+2 c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {a+c x^2} \sqrt {4 a f^2+2 c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 f^2 \sqrt {e^2-4 d f}}+\frac {c \sqrt {a+c x^2}}{f} \]
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 = 9728, normalized size = 19.61 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e x + d\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{x\,\left (f\,x^2+e\,x+d\right )} \,d x \]
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 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{x \left (d + e x + f x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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