Optimal. Leaf size=121 \[ \frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {b (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {474, 466, 396,
223, 212} \begin {gather*} -\frac {b (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}}+\frac {2 b x (b c-a d)}{d^3 \sqrt {c+d x^2}}+\frac {x^3 (b c-a d)^2}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 396
Rule 466
Rule 474
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {x^2 \left (3 b c (b c-2 a d)-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {\int \frac {-6 b c d (b c-a d)+3 b^2 c d^2 x^2}{\sqrt {c+d x^2}} \, dx}{3 c d^4}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {(b (5 b c-4 a d)) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 d^3}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {(b (5 b c-4 a d)) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 d^3}\\ &=\frac {(b c-a d)^2 x^3}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-a d) x}{d^3 \sqrt {c+d x^2}}+\frac {b^2 x \sqrt {c+d x^2}}{2 d^3}-\frac {b (5 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.21, size = 117, normalized size = 0.97 \begin {gather*} \frac {x \left (2 a^2 d^3 x^2-4 a b c d \left (3 c+4 d x^2\right )+b^2 c \left (15 c^2+20 c d x^2+3 d^2 x^4\right )\right )}{6 c d^3 \left (c+d x^2\right )^{3/2}}+\frac {b (5 b c-4 a d) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{2 d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 207, normalized size = 1.71
method | result | size |
default | \(b^{2} \left (\frac {x^{5}}{2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {5 c \left (-\frac {x^{3}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{d}\right )}{2 d}\right )+2 a b \left (-\frac {x^{3}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{d}\right )+a^{2} \left (-\frac {x}{2 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {c \left (\frac {x}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {d \,x^{2}+c}}\right )}{2 d}\right )\) | \(207\) |
risch | \(\frac {b^{2} x \sqrt {d \,x^{2}+c}}{2 d^{3}}+\frac {2 b \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a}{d^{\frac {5}{2}}}-\frac {5 b^{2} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) c}{2 d^{\frac {7}{2}}}+\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{6 d^{2} c \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {4 \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a b}{3 d^{3} \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {7 c \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{6 d^{4} \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 d^{2} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a b c}{6 d^{3} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, b^{2} c^{2}}{12 d^{4} \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{6 d^{2} c \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {4 \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, a b}{3 d^{3} \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {7 c \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}\, b^{2}}{6 d^{4} \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a^{2}}{12 d^{2} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, a b c}{6 d^{3} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}\, b^{2} c^{2}}{12 d^{4} \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}\) | \(866\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs.
\(2 (103) = 206\).
time = 0.31, size = 211, normalized size = 1.74 \begin {gather*} \frac {b^{2} x^{5}}{2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {2}{3} \, a b x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )} + \frac {5 \, b^{2} c x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{6 \, d} + \frac {5 \, b^{2} c x}{6 \, \sqrt {d x^{2} + c} d^{3}} - \frac {2 \, a b x}{3 \, \sqrt {d x^{2} + c} d^{2}} - \frac {a^{2} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {a^{2} x}{3 \, \sqrt {d x^{2} + c} c d} - \frac {5 \, b^{2} c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, d^{\frac {7}{2}}} + \frac {2 \, a b \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.75, size = 409, normalized size = 3.38 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d + {\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (3 \, b^{2} c d^{3} x^{5} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}, \frac {3 \, {\left (5 \, b^{2} c^{4} - 4 \, a b c^{3} d + {\left (5 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3}\right )} x^{4} + 2 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, b^{2} c d^{3} x^{5} + 2 \, {\left (10 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{6 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}\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} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.26, size = 130, normalized size = 1.07 \begin {gather*} \frac {{\left ({\left (\frac {3 \, b^{2} x^{2}}{d} + \frac {2 \, {\left (10 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + a^{2} d^{5}\right )}}{c d^{5}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{3} d^{2} - 4 \, a b c^{2} d^{3}\right )}}{c d^{5}}\right )} x}{6 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b^{2} c - 4 \, a b d\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{2 \, d^{\frac {7}{2}}} \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\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________