Optimal. Leaf size=499 \[ \frac {c e x \left (a \left (e^2-2 d f\right )+c d^2\right )+a f \left (a \left (e^2-d f\right )+c d^2\right )}{a f^2 \sqrt {a+c x^2} \left ((c d-a f)^2+a c e^2\right )}-\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (a \left (e^2-d f\right )+c d^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (2 a d e f-\left (\sqrt {e^2-4 d f}+e\right ) \left (a \left (e^2-d f\right )+c d^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}-\frac {1}{c f \sqrt {a+c x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 2.11, antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6728, 191, 261, 1017, 1034, 725, 206} \begin {gather*} \frac {c e x \left (a \left (e^2-2 d f\right )+c d^2\right )+a f \left (a \left (e^2-d f\right )+c d^2\right )}{a f^2 \sqrt {a+c x^2} \left ((c d-a f)^2+a c e^2\right )}-\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (a \left (e^2-d f\right )+c d^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} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (2 a d e f-\left (\sqrt {e^2-4 d f}+e\right ) \left (a \left (e^2-d f\right )+c d^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} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}-\frac {1}{c f \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 206
Rule 261
Rule 725
Rule 1017
Rule 1034
Rule 6728
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx &=\int \left (-\frac {e}{f^2 \left (a+c x^2\right )^{3/2}}+\frac {x}{f \left (a+c x^2\right )^{3/2}}+\frac {d e+\left (e^2-d f\right ) x}{f^2 \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {d e+\left (e^2-d f\right ) x}{\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx}{f^2}-\frac {e \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{f^2}+\frac {\int \frac {x}{\left (a+c x^2\right )^{3/2}} \, dx}{f}\\ &=-\frac {1}{c f \sqrt {a+c x^2}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}+\frac {a f \left (c d^2+a \left (e^2-d f\right )\right )+c e \left (c d^2+a \left (e^2-2 d f\right )\right ) x}{a f^2 \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}+\frac {\int \frac {2 a^2 c d e f^2+2 a c f^2 \left (c d^2+a e^2-a d f\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 a c f^2 \left (a c e^2+(c d-a f)^2\right )}\\ &=-\frac {1}{c f \sqrt {a+c x^2}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}+\frac {a f \left (c d^2+a \left (e^2-d f\right )\right )+c e \left (c d^2+a \left (e^2-2 d f\right )\right ) x}{a f^2 \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}+\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\right )\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{\sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right )}-\frac {\left (2 a d e f-\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\right )\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{\sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right )}\\ &=-\frac {1}{c f \sqrt {a+c x^2}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}+\frac {a f \left (c d^2+a \left (e^2-d f\right )\right )+c e \left (c d^2+a \left (e^2-2 d f\right )\right ) x}{a f^2 \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}-\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\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 )}{\sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right )}+\frac {\left (2 a d e f-\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\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 )}{\sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right )}\\ &=-\frac {1}{c f \sqrt {a+c x^2}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}+\frac {a f \left (c d^2+a \left (e^2-d f\right )\right )+c e \left (c d^2+a \left (e^2-2 d f\right )\right ) x}{a f^2 \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}-\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\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} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (2 a d e f-\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\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} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.70, size = 577, normalized size = 1.16 \begin {gather*} \frac {\left (-\frac {e \left (e^2-3 d f\right )}{\sqrt {e^2-4 d f}}-d f+e^2\right ) \left (2 a f+c x \left (e-\sqrt {e^2-4 d f}\right )\right )}{a f^2 \sqrt {a+c x^2} \left (4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )}+\frac {\left (\frac {e \left (e^2-3 d f\right )}{\sqrt {e^2-4 d f}}-d f+e^2\right ) \left (2 a f+c x \left (\sqrt {e^2-4 d f}+e\right )\right )}{a f^2 \sqrt {a+c x^2} \left (4 a f^2+c \left (\sqrt {e^2-4 d f}+e\right )^2\right )}+\frac {\sqrt {2} \left (-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}-3 d e f+e^3\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 {e^2-4 d f} \left (2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right )^{3/2}}-\frac {\sqrt {2} \left (e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}-3 d e f+e^3\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 {e^2-4 d f} \left (2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right )^{3/2}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}-\frac {1}{c f \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [C] time = 0.76, size = 425, normalized size = 0.85 \begin {gather*} \frac {\text {RootSum}\left [\text {$\#$1}^4 f-2 \text {$\#$1}^3 \sqrt {c} e-2 \text {$\#$1}^2 a f+4 \text {$\#$1}^2 c d+2 \text {$\#$1} a \sqrt {c} e+a^2 f\&,\frac {\text {$\#$1}^2 c d^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-\text {$\#$1}^2 a d f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+\text {$\#$1}^2 a e^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+a^2 d f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+a^2 \left (-e^2\right ) \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-a c d^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-2 \text {$\#$1} a \sqrt {c} d e \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 \text {$\#$1}^3 f-3 \text {$\#$1}^2 \sqrt {c} e-2 \text {$\#$1} a f+4 \text {$\#$1} c d+a \sqrt {c} e}\&\right ]}{a^2 f^2-2 a c d f+a c e^2+c^2 d^2}+\frac {a^2 (-f)+a c d-a c e x}{c \sqrt {a+c x^2} \left (a^2 f^2-2 a c d f+a c e^2+c^2 d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 6124, normalized size = 12.27 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________