Optimal. Leaf size=153 \[ -\frac {d^2 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4}-\frac {d \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} e^4}+\frac {d \sqrt {a+c x^2} (2 d-e x)}{2 e^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 c e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1654, 12, 815, 844, 217, 206, 725} \[ -\frac {d^2 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4}-\frac {d \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} e^4}+\frac {d \sqrt {a+c x^2} (2 d-e x)}{2 e^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 c e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 206
Rule 217
Rule 725
Rule 815
Rule 844
Rule 1654
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {a+c x^2}}{d+e x} \, dx &=\frac {\left (a+c x^2\right )^{3/2}}{3 c e}+\frac {\int -\frac {3 c d e x \sqrt {a+c x^2}}{d+e x} \, dx}{3 c e^2}\\ &=\frac {\left (a+c x^2\right )^{3/2}}{3 c e}-\frac {d \int \frac {x \sqrt {a+c x^2}}{d+e x} \, dx}{e}\\ &=\frac {d (2 d-e x) \sqrt {a+c x^2}}{2 e^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 c e}-\frac {d \int \frac {-a c d e+c \left (2 c d^2+a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c e^3}\\ &=\frac {d (2 d-e x) \sqrt {a+c x^2}}{2 e^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 c e}+\frac {\left (d^2 \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^4}-\frac {\left (d \left (2 c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 e^4}\\ &=\frac {d (2 d-e x) \sqrt {a+c x^2}}{2 e^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 c e}-\frac {\left (d^2 \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^4}-\frac {\left (d \left (2 c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 e^4}\\ &=\frac {d (2 d-e x) \sqrt {a+c x^2}}{2 e^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 c e}-\frac {d \left (2 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} e^4}-\frac {d^2 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.32, size = 193, normalized size = 1.26 \[ \frac {-6 c^{3/2} d^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+e \sqrt {a+c x^2} \left (2 a e^2+c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 c d^2 \sqrt {a e^2+c d^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )-\frac {3 \sqrt {a} \sqrt {c} d e^2 \sqrt {a+c x^2} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}}{6 c e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.50, size = 776, normalized size = 5.07 \[ \left [\frac {6 \, \sqrt {c d^{2} + a e^{2}} c d^{2} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 3 \, {\left (2 \, c d^{3} + a d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, c e^{4}}, -\frac {12 \, \sqrt {-c d^{2} - a e^{2}} c d^{2} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (2 \, c d^{3} + a d e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, c e^{4}}, \frac {3 \, \sqrt {c d^{2} + a e^{2}} c d^{2} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 3 \, {\left (2 \, c d^{3} + a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, c e^{4}}, -\frac {6 \, \sqrt {-c d^{2} - a e^{2}} c d^{2} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (2 \, c d^{3} + a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c e^{3} x^{2} - 3 \, c d e^{2} x + 6 \, c d^{2} e + 2 \, a e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, c e^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 157, normalized size = 1.03 \[ \frac {2 \, {\left (c d^{4} + a d^{2} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right ) e^{\left (-4\right )}}{\sqrt {-c d^{2} - a e^{2}}} + \frac {{\left (2 \, c d^{3} + a d e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, \sqrt {c}} + \frac {1}{6} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, x e^{\left (-1\right )} - 3 \, d e^{\left (-2\right )}\right )} x + \frac {2 \, {\left (3 \, c d^{2} e^{7} + a e^{9}\right )} e^{\left (-10\right )}}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 448, normalized size = 2.93 \[ -\frac {a \,d^{2} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{3}}-\frac {c \,d^{4} \ln \left (\frac {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, e^{5}}-\frac {a d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}\, e^{2}}-\frac {\sqrt {c}\, d^{3} \ln \left (\frac {-\frac {c d}{e}+\left (x +\frac {d}{e}\right ) c}{\sqrt {c}}+\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e^{4}}-\frac {\sqrt {c \,x^{2}+a}\, d x}{2 e^{2}}+\frac {\sqrt {-\frac {2 \left (x +\frac {d}{e}\right ) c d}{e}+\left (x +\frac {d}{e}\right )^{2} c +\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, d^{2}}{e^{3}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 144, normalized size = 0.94 \[ -\frac {\sqrt {c x^{2} + a} d x}{2 \, e^{2}} - \frac {\sqrt {c} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{e^{4}} - \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {c} e^{2}} + \frac {\sqrt {a + \frac {c d^{2}}{e^{2}}} d^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{e^{3}} + \frac {\sqrt {c x^{2} + a} d^{2}}{e^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{3 \, c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {a + c x^{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________