Optimal. Leaf size=73 \[ -\frac {2 a^2 A}{\sqrt {x}}+2 a^2 B \sqrt {x}+\frac {4}{3} a A c x^{3/2}+\frac {4}{5} a B c x^{5/2}+\frac {2}{7} A c^2 x^{7/2}+\frac {2}{9} B c^2 x^{9/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \begin {gather*} -\frac {2 a^2 A}{\sqrt {x}}+2 a^2 B \sqrt {x}+\frac {4}{3} a A c x^{3/2}+\frac {4}{5} a B c x^{5/2}+\frac {2}{7} A c^2 x^{7/2}+\frac {2}{9} B c^2 x^{9/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 766
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{x^{3/2}} \, dx &=\int \left (\frac {a^2 A}{x^{3/2}}+\frac {a^2 B}{\sqrt {x}}+2 a A c \sqrt {x}+2 a B c x^{3/2}+A c^2 x^{5/2}+B c^2 x^{7/2}\right ) \, dx\\ &=-\frac {2 a^2 A}{\sqrt {x}}+2 a^2 B \sqrt {x}+\frac {4}{3} a A c x^{3/2}+\frac {4}{5} a B c x^{5/2}+\frac {2}{7} A c^2 x^{7/2}+\frac {2}{9} B c^2 x^{9/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 52, normalized size = 0.71 \begin {gather*} \frac {-630 a^2 (A-B x)+84 a c x^2 (5 A+3 B x)+10 c^2 x^4 (9 A+7 B x)}{315 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.04, size = 57, normalized size = 0.78 \begin {gather*} \frac {2 \left (-315 a^2 A+315 a^2 B x+210 a A c x^2+126 a B c x^3+45 A c^2 x^4+35 B c^2 x^5\right )}{315 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 53, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (35 \, B c^{2} x^{5} + 45 \, A c^{2} x^{4} + 126 \, B a c x^{3} + 210 \, A a c x^{2} + 315 \, B a^{2} x - 315 \, A a^{2}\right )}}{315 \, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 53, normalized size = 0.73 \begin {gather*} \frac {2}{9} \, B c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, A c^{2} x^{\frac {7}{2}} + \frac {4}{5} \, B a c x^{\frac {5}{2}} + \frac {4}{3} \, A a c x^{\frac {3}{2}} + 2 \, B a^{2} \sqrt {x} - \frac {2 \, A a^{2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 54, normalized size = 0.74 \begin {gather*} -\frac {2 \left (-35 B \,c^{2} x^{5}-45 A \,c^{2} x^{4}-126 B a c \,x^{3}-210 A a c \,x^{2}-315 B \,a^{2} x +315 A \,a^{2}\right )}{315 \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.66, size = 53, normalized size = 0.73 \begin {gather*} \frac {2}{9} \, B c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, A c^{2} x^{\frac {7}{2}} + \frac {4}{5} \, B a c x^{\frac {5}{2}} + \frac {4}{3} \, A a c x^{\frac {3}{2}} + 2 \, B a^{2} \sqrt {x} - \frac {2 \, A a^{2}}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 53, normalized size = 0.73 \begin {gather*} 2\,B\,a^2\,\sqrt {x}-\frac {2\,A\,a^2}{\sqrt {x}}+\frac {2\,A\,c^2\,x^{7/2}}{7}+\frac {2\,B\,c^2\,x^{9/2}}{9}+\frac {4\,A\,a\,c\,x^{3/2}}{3}+\frac {4\,B\,a\,c\,x^{5/2}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.68, size = 76, normalized size = 1.04 \begin {gather*} - \frac {2 A a^{2}}{\sqrt {x}} + \frac {4 A a c x^{\frac {3}{2}}}{3} + \frac {2 A c^{2} x^{\frac {7}{2}}}{7} + 2 B a^{2} \sqrt {x} + \frac {4 B a c x^{\frac {5}{2}}}{5} + \frac {2 B c^{2} x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________