Optimal. Leaf size=116 \[ -\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt {x} (a+b x)}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac {2 b B \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \begin {gather*} -\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt {x} (a+b x)}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac {2 b B \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{x^{5/2}} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a A b}{x^{5/2}}+\frac {b (A b+a B)}{x^{3/2}}+\frac {b^2 B}{\sqrt {x}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 b B \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 46, normalized size = 0.40 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 9.42, size = 49, normalized size = 0.42 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-a A-3 a B x-3 A b x+3 b B x^2\right )}{3 x^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 27, normalized size = 0.23 \begin {gather*} \frac {2 \, {\left (3 \, B b x^{2} - A a - 3 \, {\left (B a + A b\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 = 0.16, size = 51, normalized size = 0.44 \begin {gather*} 2 \, B b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, {\left (3 \, B a x \mathrm {sgn}\left (b x + a\right ) + 3 \, A b x \mathrm {sgn}\left (b x + a\right ) + A a \mathrm {sgn}\left (b x + a\right )\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 = 43, normalized size = 0.37 \begin {gather*} -\frac {2 \left (-3 B b \,x^{2}+3 A b x +3 B a x +A a \right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (b x +a \right ) x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.64, size = 33, normalized size = 0.28 \begin {gather*} \frac {2 \, {\left (b x^{2} - a x\right )} B}{x^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, b x^{2} + a x\right )} A}{3 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.36, size = 54, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,A\,a}{3\,b}-2\,B\,x^2+\frac {x\,\left (6\,A\,b+6\,B\,a\right )}{3\,b}\right )}{x^{5/2}+\frac {a\,x^{3/2}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________