Optimal. Leaf size=53 \[ -\frac {2 A (a+b x)^{7/2}}{9 a x^{9/2}}+\frac {2 (2 A b-9 a B) (a+b x)^{7/2}}{63 a^2 x^{7/2}} \]
[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)^{7/2} (2 A b-9 a B)}{63 a^2 x^{7/2}}-\frac {2 A (a+b x)^{7/2}}{9 a x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{x^{11/2}} \, dx &=-\frac {2 A (a+b x)^{7/2}}{9 a x^{9/2}}+\frac {\left (2 \left (-A b+\frac {9 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{9 a}\\ &=-\frac {2 A (a+b x)^{7/2}}{9 a x^{9/2}}+\frac {2 (2 A b-9 a B) (a+b x)^{7/2}}{63 a^2 x^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 36, normalized size = 0.68 \begin {gather*} -\frac {2 (a+b x)^{7/2} (7 a A-2 A b x+9 a B x)}{63 a^2 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 77, normalized size = 1.45
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-2 A b x +9 B a x +7 A a \right )}{63 x^{\frac {9}{2}} a^{2}}\) | \(31\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-2 A \,b^{3} x^{3}+9 B a \,b^{2} x^{3}+3 a A \,b^{2} x^{2}+18 B \,a^{2} b \,x^{2}+12 a^{2} A b x +9 a^{3} B x +7 a^{3} A \right )}{63 x^{\frac {9}{2}} a^{2}}\) | \(77\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-2 A \,b^{4} x^{4}+9 B a \,b^{3} x^{4}+A a \,b^{3} x^{3}+27 B \,a^{2} b^{2} x^{3}+15 A \,a^{2} b^{2} x^{2}+27 B \,a^{3} b \,x^{2}+19 A \,a^{3} b x +9 B \,a^{4} x +7 A \,a^{4}\right )}{63 x^{\frac {9}{2}} a^{2}}\) | \(100\) |
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. 258 vs.
\(2 (41) = 82\).
time = 0.29, size = 258, normalized size = 4.87 \begin {gather*} -\frac {2 \, \sqrt {b x^{2} + a x} B b^{3}}{7 \, a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{4}}{63 \, a^{2} x} + \frac {\sqrt {b x^{2} + a x} B b^{2}}{7 \, x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{63 \, a x^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} B a b}{28 \, x^{3}} + \frac {\sqrt {b x^{2} + a x} A b^{2}}{42 \, x^{3}} - \frac {15 \, \sqrt {b x^{2} + a x} B a^{2}}{28 \, x^{4}} - \frac {5 \, \sqrt {b x^{2} + a x} A a b}{252 \, x^{4}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{4 \, x^{5}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{36 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{x^{6}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{12 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{2 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (41) = 82\).
time = 1.10, size = 98, normalized size = 1.85 \begin {gather*} -\frac {2 \, {\left (7 \, A a^{4} + {\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} + {\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{3} + 3 \, {\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{2} + {\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{63 \, a^{2} x^{\frac {9}{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. 209 vs.
\(2 (49) = 98\).
time = 154.49, size = 209, normalized size = 3.94 \begin {gather*} A \left (- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{9 x^{4}} - \frac {38 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{63 x^{3}} - \frac {10 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{21 x^{2}} - \frac {2 b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{63 a x} + \frac {4 b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{63 a^{2}}\right ) + B \left (- \frac {2 a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{7 x^{3}} - \frac {6 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{7 x^{2}} - \frac {6 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{7 x} - \frac {2 b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{7 a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.71, size = 80, normalized size = 1.51 \begin {gather*} -\frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} b {\left (\frac {{\left (9 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )} {\left (b x + a\right )}}{a^{4}} - \frac {9 \, {\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{4}}\right )}}{63 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.86, size = 95, normalized size = 1.79 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{9}+\frac {x\,\left (18\,B\,a^4+38\,A\,b\,a^3\right )}{63\,a^2}-\frac {x^4\,\left (4\,A\,b^4-18\,B\,a\,b^3\right )}{63\,a^2}+\frac {2\,b\,x^2\,\left (5\,A\,b+9\,B\,a\right )}{21}+\frac {2\,b^2\,x^3\,\left (A\,b+27\,B\,a\right )}{63\,a}\right )}{x^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________