Optimal. Leaf size=153 \[ \frac {d (2 d-e x) \sqrt {a+c x^2}}{2 e^3}+\frac {\left (a+c x^2\right )^{3/2}}{3 c e}-\frac {d \left (2 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c} e^4}-\frac {d^2 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, 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 = {1668, 12, 829,
858, 223, 212, 739} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 212
Rule 223
Rule 739
Rule 829
Rule 858
Rule 1668
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 ) \text {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 ) \text {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 = 0.40, size = 161, normalized size = 1.05 \begin {gather*} \frac {e \sqrt {a+c x^2} \left (6 c d^2+2 a e^2-3 c d e x+2 c e^2 x^2\right )+12 c d^2 \sqrt {-c d^2-a e^2} \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )+3 \sqrt {c} d \left (2 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 c e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs.
\(2(131)=262\).
time = 0.08, size = 323, normalized size = 2.11
method | result | size |
default | \(\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c e}-\frac {d \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{e^{2}}+\frac {d^{2} \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\) | \(323\) |
risch | \(\frac {\left (2 c \,e^{2} x^{2}-3 c d e x +2 a \,e^{2}+6 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c \,e^{3}}-\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a}{2 e^{2} \sqrt {c}}-\frac {d^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) \sqrt {c}}{e^{4}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right ) a}{e^{3} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {d^{4} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right ) c}{e^{5} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 140, normalized size = 0.92 \begin {gather*} -\sqrt {c} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )} + \sqrt {c d^{2} e^{\left (-2\right )} + a} d^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-3\right )} - \frac {1}{2} \, \sqrt {c x^{2} + a} d x e^{\left (-2\right )} - \frac {a d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{2 \, \sqrt {c}} + \sqrt {c x^{2} + a} d^{2} e^{\left (-3\right )} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.97, size = 745, normalized size = 4.87 \begin {gather*} \left [\frac {{\left (6 \, \sqrt {c d^{2} + a e^{2}} c d^{2} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + 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 (3 \, c d x e^{2} - 6 \, c d^{2} e - 2 \, {\left (c x^{2} + a\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-4\right )}}{12 \, c}, \frac {{\left (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}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{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 (3 \, c d x e^{2} - 6 \, c d^{2} e - 2 \, {\left (c x^{2} + a\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-4\right )}}{12 \, c}, \frac {{\left (3 \, \sqrt {c d^{2} + a e^{2}} c d^{2} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + 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 (3 \, c d x e^{2} - 6 \, c d^{2} e - 2 \, {\left (c x^{2} + a\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-4\right )}}{6 \, c}, \frac {{\left (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}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{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 (3 \, c d x e^{2} - 6 \, c d^{2} e - 2 \, {\left (c x^{2} + a\right )} e^{3}\right )} \sqrt {c x^{2} + a}\right )} e^{\left (-4\right )}}{6 \, c}\right ] \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^{2} \sqrt {a + c x^{2}}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.86, size = 157, normalized size = 1.03 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________