Optimal. Leaf size=192 \[ -\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223,
212} \begin {gather*} \frac {3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{7/2}}-\frac {3 a^3 \sqrt {x} \sqrt {a+b x} (2 A b-a B)}{128 b^3}+\frac {a^2 x^{3/2} \sqrt {a+b x} (2 A b-a B)}{64 b^2}+\frac {a x^{5/2} \sqrt {a+b x} (2 A b-a B)}{16 b}+\frac {x^{5/2} (a+b x)^{3/2} (2 A b-a B)}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps
\begin {align*} \int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx &=\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (5 A b-\frac {5 a B}{2}\right ) \int x^{3/2} (a+b x)^{3/2} \, dx}{5 b}\\ &=\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {(3 a (2 A b-a B)) \int x^{3/2} \sqrt {a+b x} \, dx}{16 b}\\ &=\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (a^2 (2 A b-a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{32 b}\\ &=\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}-\frac {\left (3 a^3 (2 A b-a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b^2}\\ &=-\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (3 a^4 (2 A b-a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^3}\\ &=-\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (3 a^4 (2 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^3}\\ &=-\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {\left (3 a^4 (2 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^3}\\ &=-\frac {3 a^3 (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{128 b^3}+\frac {a^2 (2 A b-a B) x^{3/2} \sqrt {a+b x}}{64 b^2}+\frac {a (2 A b-a B) x^{5/2} \sqrt {a+b x}}{16 b}+\frac {(2 A b-a B) x^{5/2} (a+b x)^{3/2}}{8 b}+\frac {B x^{5/2} (a+b x)^{5/2}}{5 b}+\frac {3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.21, size = 136, normalized size = 0.71 \begin {gather*} \frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (15 a^4 B-10 a^3 b (3 A+B x)+4 a^2 b^2 x (5 A+2 B x)+32 b^4 x^3 (5 A+4 B x)+16 a b^3 x^2 (15 A+11 B x)\right )+15 a^4 (-2 A b+a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{640 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 260, normalized size = 1.35
method | result | size |
risch | \(-\frac {\left (-128 B \,x^{4} b^{4}-160 A \,b^{4} x^{3}-176 B a \,b^{3} x^{3}-240 A a \,b^{3} x^{2}-8 B \,a^{2} b^{2} x^{2}-20 A \,a^{2} b^{2} x +10 B \,a^{3} b x +30 A \,a^{3} b -15 a^{4} B \right ) \sqrt {b x +a}\, \sqrt {x}}{640 b^{3}}+\frac {\left (\frac {3 a^{4} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) A}{128 b^{\frac {5}{2}}}-\frac {3 a^{5} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) B}{256 b^{\frac {7}{2}}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\) | \(186\) |
default | \(\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (256 B \,b^{\frac {9}{2}} x^{4} \sqrt {\left (b x +a \right ) x}+320 A \,b^{\frac {9}{2}} x^{3} \sqrt {\left (b x +a \right ) x}+352 B a \,b^{\frac {7}{2}} x^{3} \sqrt {\left (b x +a \right ) x}+480 A a \,b^{\frac {7}{2}} x^{2} \sqrt {\left (b x +a \right ) x}+16 B \,a^{2} b^{\frac {5}{2}} x^{2} \sqrt {\left (b x +a \right ) x}+40 A \,b^{\frac {5}{2}} \sqrt {\left (b x +a \right ) x}\, a^{2} x -20 B \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, a^{3} x +30 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4} b -60 A \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}\, a^{3}-15 B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5}+30 B \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\, a^{4}\right )}{1280 b^{\frac {7}{2}} \sqrt {\left (b x +a \right ) x}}\) | \(260\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 236, normalized size = 1.23 \begin {gather*} \frac {1}{4} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A x + \frac {3 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a x}{8 \, b} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{2} x}{32 \, b} - \frac {3 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} + \frac {3 \, A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {5}{2}}} + \frac {3 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2}}{16 \, b^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{3}}{64 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{5 \, b} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.96, size = 290, normalized size = 1.51 \begin {gather*} \left [-\frac {15 \, {\left (B a^{5} - 2 \, A a^{4} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (128 \, B b^{5} x^{4} + 15 \, B a^{4} b - 30 \, A a^{3} b^{2} + 16 \, {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} x^{3} + 8 \, {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} x^{2} - 10 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1280 \, b^{4}}, \frac {15 \, {\left (B a^{5} - 2 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (128 \, B b^{5} x^{4} + 15 \, B a^{4} b - 30 \, A a^{3} b^{2} + 16 \, {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} x^{3} + 8 \, {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} x^{2} - 10 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{640 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 211.84, size = 1856, normalized size = 9.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
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]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{3/2}\,\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________