Optimal. Leaf size=84 \[ -\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}}+\frac {2 (4 A b-9 a B) (a+b x)^{5/2}}{63 a^2 x^{7/2}}-\frac {4 b (4 A b-9 a B) (a+b x)^{5/2}}{315 a^3 x^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37}
\begin {gather*} -\frac {4 b (a+b x)^{5/2} (4 A b-9 a B)}{315 a^3 x^{5/2}}+\frac {2 (a+b x)^{5/2} (4 A b-9 a B)}{63 a^2 x^{7/2}}-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^{11/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}}+\frac {\left (2 \left (-2 A b+\frac {9 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{9/2}} \, dx}{9 a}\\ &=-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}}+\frac {2 (4 A b-9 a B) (a+b x)^{5/2}}{63 a^2 x^{7/2}}+\frac {(2 b (4 A b-9 a B)) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{63 a^2}\\ &=-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}}+\frac {2 (4 A b-9 a B) (a+b x)^{5/2}}{63 a^2 x^{7/2}}-\frac {4 b (4 A b-9 a B) (a+b x)^{5/2}}{315 a^3 x^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 58, normalized size = 0.69 \begin {gather*} -\frac {2 (a+b x)^{5/2} \left (35 a^2 A-20 a A b x+45 a^2 B x+8 A b^2 x^2-18 a b B x^2\right )}{315 a^3 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 77, normalized size = 0.92
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (8 A \,b^{2} x^{2}-18 B a b \,x^{2}-20 a A b x +45 a^{2} B x +35 a^{2} A \right )}{315 x^{\frac {9}{2}} a^{3}}\) | \(53\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (8 A \,b^{3} x^{3}-18 B a \,b^{2} x^{3}-12 a A \,b^{2} x^{2}+27 B \,a^{2} b \,x^{2}+15 a^{2} A b x +45 a^{3} B x +35 a^{3} A \right )}{315 x^{\frac {9}{2}} a^{3}}\) | \(77\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{4} x^{4}-18 B a \,b^{3} x^{4}-4 A a \,b^{3} x^{3}+9 B \,a^{2} b^{2} x^{3}+3 A \,a^{2} b^{2} x^{2}+72 B \,a^{3} b \,x^{2}+50 A \,a^{3} b x +45 B \,a^{4} x +35 A \,a^{4}\right )}{315 x^{\frac {9}{2}} a^{3}}\) | \(101\) |
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. 222 vs.
\(2 (66) = 132\).
time = 0.29, size = 222, normalized size = 2.64 \begin {gather*} \frac {4 \, \sqrt {b x^{2} + a x} B b^{3}}{35 \, a^{2} x} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{315 \, a^{3} x} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{35 \, a x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} A b^{3}}{315 \, a^{2} x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} B b}{70 \, x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{105 \, a x^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{14 \, x^{4}} + \frac {\sqrt {b x^{2} + a x} A b}{63 \, x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{2 \, x^{5}} + \frac {\sqrt {b x^{2} + a x} A a}{9 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{3 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.84, size = 100, normalized size = 1.19 \begin {gather*} -\frac {2 \, {\left (35 \, A a^{4} - 2 \, {\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} + {\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \, {\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{315 \, a^{3} 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. 500 vs.
\(2 (82) = 164\).
time = 145.21, size = 500, normalized size = 5.95 \begin {gather*} A \left (- \frac {70 a^{6} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {240 a^{5} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {276 a^{4} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {104 a^{3} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {6 a^{2} b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {24 a b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}} - \frac {16 b^{\frac {21}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{315 a^{5} b^{4} x^{4} + 630 a^{4} b^{5} x^{5} + 315 a^{3} b^{6} x^{6}}\right ) + B \left (- \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{7 x^{3}} - \frac {16 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{35 x^{2}} - \frac {2 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a x} + \frac {4 b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.12, size = 110, normalized size = 1.31 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (9 \, B a^{2} b^{8} - 4 \, A a b^{9}\right )} {\left (b x + a\right )}}{a^{4}} - \frac {9 \, {\left (9 \, B a^{3} b^{8} - 4 \, A a^{2} b^{9}\right )}}{a^{4}}\right )} + \frac {63 \, {\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{4}}\right )} b}{315 \, {\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 = 0.79, size = 96, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{9}+\frac {x\,\left (90\,B\,a^4+100\,A\,b\,a^3\right )}{315\,a^3}+\frac {x^4\,\left (16\,A\,b^4-36\,B\,a\,b^3\right )}{315\,a^3}-\frac {2\,b^2\,x^3\,\left (4\,A\,b-9\,B\,a\right )}{315\,a^2}+\frac {2\,b\,x^2\,\left (A\,b+24\,B\,a\right )}{105\,a}\right )}{x^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________