Optimal. Leaf size=53 \[ -\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}}+\frac {2 (2 A b-5 a B) (a+b x)^{3/2}}{15 a^2 x^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37}
\begin {gather*} \frac {2 (a+b x)^{3/2} (2 A b-5 a B)}{15 a^2 x^{3/2}}-\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 79
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x^{7/2}} \, dx &=-\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}}+\frac {\left (2 \left (-A b+\frac {5 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx}{5 a}\\ &=-\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}}+\frac {2 (2 A b-5 a B) (a+b x)^{3/2}}{15 a^2 x^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 36, normalized size = 0.68 \begin {gather*} -\frac {2 (a+b x)^{3/2} (3 a A-2 A b x+5 a B x)}{15 a^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 31, normalized size = 0.58
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-2 A b x +5 B a x +3 A a \right )}{15 x^{\frac {5}{2}} a^{2}}\) | \(31\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-2 A b x +5 B a x +3 A a \right )}{15 x^{\frac {5}{2}} a^{2}}\) | \(31\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-2 A \,b^{2} x^{2}+5 B a b \,x^{2}+a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 x^{\frac {5}{2}} a^{2}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (41) = 82\).
time = 0.29, size = 100, normalized size = 1.89 \begin {gather*} -\frac {2 \, \sqrt {b x^{2} + a x} B b}{3 \, a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{2}}{15 \, a^{2} x} - \frac {2 \, \sqrt {b x^{2} + a x} B}{3 \, x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{15 \, a x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{5 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.94, size = 51, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (3 \, A a^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} x^{2} + {\left (5 \, B a^{2} + A a b\right )} x\right )} \sqrt {b x + a}}{15 \, a^{2} x^{\frac {5}{2}}} \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. 110 vs.
\(2 (49) = 98\).
time = 26.75, size = 110, normalized size = 2.08 \begin {gather*} A \left (- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a x} + \frac {4 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{2}}\right ) + B \left (- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.89, size = 73, normalized size = 1.38 \begin {gather*} -\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} b {\left (\frac {{\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} {\left (b x + a\right )}}{a^{2}} - \frac {5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )}}{a^{2}}\right )}}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.67, size = 54, normalized size = 1.02 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{5}-\frac {x^2\,\left (4\,A\,b^2-10\,B\,a\,b\right )}{15\,a^2}+\frac {x\,\left (10\,B\,a^2+2\,A\,b\,a\right )}{15\,a^2}\right )}{x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________