Optimal. Leaf size=35 \[ -\frac {2 (A c+b B)}{\sqrt {x}}-\frac {2 A b}{3 x^{3/2}}+2 B c \sqrt {x} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 35, 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 = {765} \begin {gather*} -\frac {2 (A c+b B)}{\sqrt {x}}-\frac {2 A b}{3 x^{3/2}}+2 B c \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{7/2}} \, dx &=\int \left (\frac {A b}{x^{5/2}}+\frac {b B+A c}{x^{3/2}}+\frac {B c}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 A b}{3 x^{3/2}}-\frac {2 (b B+A c)}{\sqrt {x}}+2 B c \sqrt {x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 0.80 \begin {gather*} -\frac {2 (A (b+3 c x)+3 B x (b-c x))}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 31, normalized size = 0.89 \begin {gather*} \frac {2 \left (-A b-3 A c x-3 b B x+3 B c x^2\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 27, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (3 \, B c x^{2} - A b - 3 \, {\left (B b + A c\right )} x\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 = 27, normalized size = 0.77 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (3 \, B b x + 3 \, A c x + A b\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 = 27, normalized size = 0.77 \begin {gather*} -\frac {2 \left (-3 B c \,x^{2}+3 A c x +3 B b x +A b \right )}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.59, size = 27, normalized size = 0.77 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (A b + 3 \, {\left (B b + A c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.02, size = 27, normalized size = 0.77 \begin {gather*} -\frac {2\,A\,b+6\,A\,c\,x+6\,B\,b\,x-6\,B\,c\,x^2}{3\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.34, size = 41, normalized size = 1.17 \begin {gather*} - \frac {2 A b}{3 x^{\frac {3}{2}}} - \frac {2 A c}{\sqrt {x}} - \frac {2 B b}{\sqrt {x}} + 2 B c \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________