Optimal. Leaf size=110 \[ \frac {c (b c-a d)^2}{3 d^4 \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c-a d)}{d^4 \sqrt {c+d x^2}}-\frac {b (3 b c-2 a d) \sqrt {c+d x^2}}{d^4}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {457, 78}
\begin {gather*} -\frac {b \sqrt {c+d x^2} (3 b c-2 a d)}{d^4}-\frac {(b c-a d) (3 b c-a d)}{d^4 \sqrt {c+d x^2}}+\frac {c (b c-a d)^2}{3 d^4 \left (c+d x^2\right )^{3/2}}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x (a+b x)^2}{(c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {c (b c-a d)^2}{d^3 (c+d x)^{5/2}}+\frac {(b c-a d) (3 b c-a d)}{d^3 (c+d x)^{3/2}}-\frac {b (3 b c-2 a d)}{d^3 \sqrt {c+d x}}+\frac {b^2 \sqrt {c+d x}}{d^3}\right ) \, dx,x,x^2\right )\\ &=\frac {c (b c-a d)^2}{3 d^4 \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c-a d)}{d^4 \sqrt {c+d x^2}}-\frac {b (3 b c-2 a d) \sqrt {c+d x^2}}{d^4}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 98, normalized size = 0.89 \begin {gather*} \frac {-a^2 d^2 \left (2 c+3 d x^2\right )+2 a b d \left (8 c^2+12 c d x^2+3 d^2 x^4\right )+b^2 \left (-16 c^3-24 c^2 d x^2-6 c d^2 x^4+d^3 x^6\right )}{3 d^4 \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 183, normalized size = 1.66
method | result | size |
risch | \(\frac {b \left (b d \,x^{2}+6 a d -8 b c \right ) \sqrt {d \,x^{2}+c}}{3 d^{4}}-\frac {\left (a d -b c \right ) \left (3 a \,d^{2} x^{2}-9 b c d \,x^{2}+2 a c d -8 b \,c^{2}\right ) \sqrt {d \,x^{2}+c}}{3 d^{4} \left (d^{2} x^{4}+2 x^{2} d c +c^{2}\right )}\) | \(103\) |
gosper | \(-\frac {-b^{2} x^{6} d^{3}-6 a b \,d^{3} x^{4}+6 b^{2} c \,d^{2} x^{4}+3 a^{2} d^{3} x^{2}-24 a b c \,d^{2} x^{2}+24 b^{2} c^{2} d \,x^{2}+2 a^{2} c \,d^{2}-16 a b \,c^{2} d +16 b^{2} c^{3}}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{4}}\) | \(108\) |
trager | \(-\frac {-b^{2} x^{6} d^{3}-6 a b \,d^{3} x^{4}+6 b^{2} c \,d^{2} x^{4}+3 a^{2} d^{3} x^{2}-24 a b c \,d^{2} x^{2}+24 b^{2} c^{2} d \,x^{2}+2 a^{2} c \,d^{2}-16 a b \,c^{2} d +16 b^{2} c^{3}}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{4}}\) | \(108\) |
default | \(b^{2} \left (\frac {x^{6}}{3 d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 c \left (\frac {x^{4}}{d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {4 c \left (-\frac {x^{2}}{d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 c}{3 d^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\right )}{d}\right )}{d}\right )+2 a b \left (\frac {x^{4}}{d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {4 c \left (-\frac {x^{2}}{d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 c}{3 d^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\right )}{d}\right )+a^{2} \left (-\frac {x^{2}}{d \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 c}{3 d^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\right )\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 181, normalized size = 1.65 \begin {gather*} \frac {b^{2} x^{6}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {2 \, b^{2} c x^{4}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}} + \frac {2 \, a b x^{4}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {8 \, b^{2} c^{2} x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{3}} + \frac {8 \, a b c x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}} - \frac {a^{2} x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} - \frac {16 \, b^{2} c^{3}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{4}} + \frac {16 \, a b c^{2}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{3}} - \frac {2 \, a^{2} c}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.85, size = 124, normalized size = 1.13 \begin {gather*} \frac {{\left (b^{2} d^{3} x^{6} - 16 \, b^{2} c^{3} + 16 \, a b c^{2} d - 2 \, a^{2} c d^{2} - 6 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{4} - 3 \, {\left (8 \, b^{2} c^{2} d - 8 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3 \, {\left (d^{6} x^{4} + 2 \, c d^{5} x^{2} + c^{2} d^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 454 vs.
\(2 (97) = 194\).
time = 0.52, size = 454, normalized size = 4.13 \begin {gather*} \begin {cases} - \frac {2 a^{2} c d^{2}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} - \frac {3 a^{2} d^{3} x^{2}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} + \frac {16 a b c^{2} d}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} + \frac {24 a b c d^{2} x^{2}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} + \frac {6 a b d^{3} x^{4}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} - \frac {16 b^{2} c^{3}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} - \frac {24 b^{2} c^{2} d x^{2}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} - \frac {6 b^{2} c d^{2} x^{4}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} + \frac {b^{2} d^{3} x^{6}}{3 c d^{4} \sqrt {c + d x^{2}} + 3 d^{5} x^{2} \sqrt {c + d x^{2}}} & \text {for}\: d \neq 0 \\\frac {\frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.03, size = 140, normalized size = 1.27 \begin {gather*} -\frac {9 \, {\left (d x^{2} + c\right )} b^{2} c^{2} - b^{2} c^{3} - 12 \, {\left (d x^{2} + c\right )} a b c d + 2 \, a b c^{2} d + 3 \, {\left (d x^{2} + c\right )} a^{2} d^{2} - a^{2} c d^{2}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{4}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{8} - 9 \, \sqrt {d x^{2} + c} b^{2} c d^{8} + 6 \, \sqrt {d x^{2} + c} a b d^{9}}{3 \, d^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.44, size = 107, normalized size = 0.97 \begin {gather*} -\frac {2\,a^2\,c\,d^2+3\,a^2\,d^3\,x^2-16\,a\,b\,c^2\,d-24\,a\,b\,c\,d^2\,x^2-6\,a\,b\,d^3\,x^4+16\,b^2\,c^3+24\,b^2\,c^2\,d\,x^2+6\,b^2\,c\,d^2\,x^4-b^2\,d^3\,x^6}{3\,d^4\,{\left (d\,x^2+c\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________