Optimal. Leaf size=553 \[ -\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f^4 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (\sqrt {e^2-4 d f}+e\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} f^4 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )}{2 f^4}+\frac {\sqrt {a+c x^2} \left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right )}{2 f^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 f} \]
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Rubi [A] time = 2.43, antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1020, 1068, 1080, 217, 206, 1034, 725} \begin {gather*} -\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\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} f^4 \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 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (\sqrt {e^2-4 d f}+e\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (d^2 f^2-3 d e^2 f+e^4\right )\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} f^4 \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 {\sqrt {a+c x^2} \left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right )}{2 f^3}-\frac {\sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )}{2 f^4}+\frac {\left (a+c x^2\right )^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 725
Rule 1020
Rule 1034
Rule 1068
Rule 1080
Rubi steps
\begin {align*} \int \frac {x \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=\frac {\left (a+c x^2\right )^{3/2}}{3 f}+\frac {\int \frac {\sqrt {a+c x^2} \left (-3 (c d-a f) x-3 c e x^2\right )}{d+e x+f x^2} \, dx}{3 f}\\ &=\frac {\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt {a+c x^2}}{2 f^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 f}-\frac {\int \frac {-3 a c^2 d e f+3 c \left (a c e^2 f+2 (c d-a f) \left (c e^2-c d f+a f^2\right )\right ) x+3 c^2 e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right ) x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c f^3}\\ &=\frac {\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt {a+c x^2}}{2 f^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 f}-\frac {\int \frac {-3 a c^2 d e f^2-3 c^2 d e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )+\left (-3 c^2 e^2 \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )+3 c f \left (a c e^2 f+2 (c d-a f) \left (c e^2-c d f+a f^2\right )\right )\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c f^4}-\frac {\left (c e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 f^4}\\ &=\frac {\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt {a+c x^2}}{2 f^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 f}-\frac {\left (c e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 f^4}+\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{f^4 \sqrt {e^2-4 d f}}-\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{f^4 \sqrt {e^2-4 d f}}\\ &=\frac {\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt {a+c x^2}}{2 f^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 f}-\frac {\sqrt {c} e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 f^4}-\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\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 )}{f^4 \sqrt {e^2-4 d f}}+\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\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 )}{f^4 \sqrt {e^2-4 d f}}\\ &=\frac {\left (2 \left (a f^2+c \left (e^2-d f\right )\right )-c e f x\right ) \sqrt {a+c x^2}}{2 f^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 f}-\frac {\sqrt {c} e \left (3 a f^2+2 c \left (e^2-2 d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 f^4}-\frac {\left (2 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\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} f^4 \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 c d e f \left (2 a f^2+c \left (e^2-2 d f\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (a^2 f^4+2 a c f^2 \left (e^2-d f\right )+c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\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} f^4 \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 )}}\\ \end {align*}
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Mathematica [A] time = 2.01, size = 755, normalized size = 1.37 \begin {gather*} \frac {8 f^3 \left (a+c x^2\right )^{5/2} \sqrt {\frac {c x^2}{a}+1} \left (\sqrt {e^2-4 d f}-e\right )+8 f^3 \left (a+c x^2\right )^{5/2} \sqrt {\frac {c x^2}{a}+1} \left (\sqrt {e^2-4 d f}+e\right )+3 \left (e-\sqrt {e^2-4 d f}\right ) \left (2 \sqrt {c} f^2 \sqrt {a+c x^2} \left (e-\sqrt {e^2-4 d f}\right ) \left (a \sqrt {c} x \left (\frac {c x^2}{a}+1\right )^{3/2}+\sqrt {a} \left (a+c x^2\right ) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )\right )-a \left (\frac {c x^2}{a}+1\right )^{3/2} \left (4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2\right ) \left (-\sqrt {4 a f^2-2 c e \sqrt {e^2-4 d f}-4 c d f+2 c e^2} \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 )+\sqrt {c} \left (\sqrt {e^2-4 d f}-e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+2 f \sqrt {a+c x^2}\right )\right )-3 \left (\sqrt {e^2-4 d f}+e\right ) \left (2 \sqrt {c} f^2 \sqrt {a+c x^2} \left (\sqrt {e^2-4 d f}+e\right ) \left (a \sqrt {c} x \left (\frac {c x^2}{a}+1\right )^{3/2}+\sqrt {a} \left (a+c x^2\right ) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )\right )-a \left (\frac {c x^2}{a}+1\right )^{3/2} \left (4 a f^2+c \left (\sqrt {e^2-4 d f}+e\right )^2\right ) \left (-\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 )-\sqrt {c} \left (\sqrt {e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+2 f \sqrt {a+c x^2}\right )\right )}{48 a f^4 \left (\frac {c x^2}{a}+1\right )^{3/2} \sqrt {e^2-4 d f}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.89, size = 771, normalized size = 1.39 \begin {gather*} -\frac {\text {RootSum}\left [\text {$\#$1}^4 f-2 \text {$\#$1}^3 \sqrt {c} e-2 \text {$\#$1}^2 a f+4 \text {$\#$1}^2 c d+2 \text {$\#$1} a \sqrt {c} e+a^2 f\&,\frac {-\text {$\#$1}^2 a^2 f^4 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-\text {$\#$1}^2 c^2 d^2 f^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+3 \text {$\#$1}^2 c^2 d e^2 f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+\text {$\#$1}^2 \left (-c^2\right ) e^4 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+2 \text {$\#$1}^2 a c d f^3 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-2 \text {$\#$1}^2 a c e^2 f^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+a^3 f^4 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-2 a^2 c d f^3 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+2 a^2 c e^2 f^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-4 \text {$\#$1} c^{5/2} d^2 e f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+2 \text {$\#$1} c^{5/2} d e^3 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+4 \text {$\#$1} a c^{3/2} d e f^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+a c^2 d^2 f^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-3 a c^2 d e^2 f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+a c^2 e^4 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 \text {$\#$1}^3 f-3 \text {$\#$1}^2 \sqrt {c} e-2 \text {$\#$1} a f+4 \text {$\#$1} c d+a \sqrt {c} e}\&\right ]}{f^4}+\frac {\log \left (\sqrt {a+c x^2}-\sqrt {c} x\right ) \left (3 a \sqrt {c} e f^2-4 c^{3/2} d e f+2 c^{3/2} e^3\right )}{2 f^4}+\frac {\sqrt {a+c x^2} \left (8 a f^2-6 c d f+6 c e^2-3 c e f x+2 c f^2 x^2\right )}{6 f^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \mathit {sage}_{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 14709, normalized size = 26.60 \begin {gather*} \text {output too large to display} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
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 \begin {gather*} \int \frac {x\,{\left (c\,x^2+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x \left (a + c x^{2}\right )^{\frac {3}{2}}}{d + e x + f x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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