Optimal. Leaf size=41 \[ -\frac {2 a A}{3 x^{3/2}}-\frac {2 a B}{\sqrt {x}}+2 A c \sqrt {x}+\frac {2}{3} B c x^{3/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \begin {gather*} -\frac {2 a A}{3 x^{3/2}}-\frac {2 a B}{\sqrt {x}}+2 A c \sqrt {x}+\frac {2}{3} B c x^{3/2} \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 )}{x^{5/2}} \, dx &=\int \left (\frac {a A}{x^{5/2}}+\frac {a B}{x^{3/2}}+\frac {A c}{\sqrt {x}}+B c \sqrt {x}\right ) \, dx\\ &=-\frac {2 a A}{3 x^{3/2}}-\frac {2 a B}{\sqrt {x}}+2 A c \sqrt {x}+\frac {2}{3} B c x^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 32, normalized size = 0.78 \begin {gather*} \frac {2 c x^2 (3 A+B x)-2 a (A+3 B x)}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 32, normalized size = 0.78 \begin {gather*} \frac {2 \left (-a A-3 a B x+3 A c x^2+B c x^3\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 28, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (B c x^{3} + 3 \, A c x^{2} - 3 \, B a x - A a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 29, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, A c \sqrt {x} - \frac {2 \, {\left (3 \, B a x + A a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 29, normalized size = 0.71 \begin {gather*} -\frac {2 \left (-B c \,x^{3}-3 A c \,x^{2}+3 B a x +a A \right )}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.71, size = 29, normalized size = 0.71 \begin {gather*} \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, A c \sqrt {x} - \frac {2 \, {\left (3 \, B a x + A a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.05, size = 29, normalized size = 0.71 \begin {gather*} -\frac {-2\,B\,c\,x^3-6\,A\,c\,x^2+6\,B\,a\,x+2\,A\,a}{3\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.85, size = 42, normalized size = 1.02 \begin {gather*} - \frac {2 A a}{3 x^{\frac {3}{2}}} + 2 A c \sqrt {x} - \frac {2 B a}{\sqrt {x}} + \frac {2 B c x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________