Integrand size = 20, antiderivative size = 159 \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=\frac {a^2 (8 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a (8 A b-3 a B) x^{3/2} \sqrt {a+b x}}{32 b}+\frac {(8 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b}-\frac {a^3 (8 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{5/2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=-\frac {a^3 (8 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{5/2}}+\frac {a^2 \sqrt {x} \sqrt {a+b x} (8 A b-3 a B)}{64 b^2}+\frac {a x^{3/2} \sqrt {a+b x} (8 A b-3 a B)}{32 b}+\frac {x^{3/2} (a+b x)^{3/2} (8 A b-3 a B)}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b} \]
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {B x^{3/2} (a+b x)^{5/2}}{4 b}+\frac {\left (4 A b-\frac {3 a B}{2}\right ) \int \sqrt {x} (a+b x)^{3/2} \, dx}{4 b} \\ & = \frac {(8 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b}+\frac {(a (8 A b-3 a B)) \int \sqrt {x} \sqrt {a+b x} \, dx}{16 b} \\ & = \frac {a (8 A b-3 a B) x^{3/2} \sqrt {a+b x}}{32 b}+\frac {(8 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b}+\frac {\left (a^2 (8 A b-3 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{64 b} \\ & = \frac {a^2 (8 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a (8 A b-3 a B) x^{3/2} \sqrt {a+b x}}{32 b}+\frac {(8 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b}-\frac {\left (a^3 (8 A b-3 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b^2} \\ & = \frac {a^2 (8 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a (8 A b-3 a B) x^{3/2} \sqrt {a+b x}}{32 b}+\frac {(8 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b}-\frac {\left (a^3 (8 A b-3 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^2} \\ & = \frac {a^2 (8 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a (8 A b-3 a B) x^{3/2} \sqrt {a+b x}}{32 b}+\frac {(8 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b}-\frac {\left (a^3 (8 A b-3 a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^2} \\ & = \frac {a^2 (8 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{64 b^2}+\frac {a (8 A b-3 a B) x^{3/2} \sqrt {a+b x}}{32 b}+\frac {(8 A b-3 a B) x^{3/2} (a+b x)^{3/2}}{24 b}+\frac {B x^{3/2} (a+b x)^{5/2}}{4 b}-\frac {a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{5/2}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.79 \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (-9 a^3 B+6 a^2 b (4 A+B x)+16 b^3 x^2 (4 A+3 B x)+8 a b^2 x (14 A+9 B x)\right )+6 a^3 (-8 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{192 b^{5/2}} \]
[In]
[Out]
Time = 1.42 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {\left (48 b^{3} B \,x^{3}+64 A \,b^{3} x^{2}+72 B a \,b^{2} x^{2}+112 a \,b^{2} A x +6 a^{2} b B x +24 a^{2} b A -9 a^{3} B \right ) \sqrt {x}\, \sqrt {b x +a}}{192 b^{2}}-\frac {a^{3} \left (8 A b -3 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{128 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(135\) |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {x}\, \left (-96 B \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}-128 A \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}-144 B a \,b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}-224 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a x -12 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2} x +24 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b -48 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2}-9 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4}+18 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{3}\right )}{384 b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}}\) | \(218\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.57 \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=\left [-\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (48 \, B b^{4} x^{3} - 9 \, B a^{3} b + 24 \, A a^{2} b^{2} + 8 \, {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{3}}, -\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (48 \, B b^{4} x^{3} - 9 \, B a^{3} b + 24 \, A a^{2} b^{2} + 8 \, {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{3}}\right ] \]
[In]
[Out]
Time = 1.41 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.87 \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=2 A a \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x} + 2 b \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {\sqrt {x} \log {\left (\sqrt {x} \right )}}{\sqrt {b x}} & \text {otherwise} \end {cases}\right )}{8 b} + \sqrt {a + b x} \left (\frac {a \sqrt {x}}{8 b} + \frac {x^{\frac {3}{2}}}{4}\right ) & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{\frac {3}{2}}}{3} & \text {otherwise} \end {cases}\right ) + 2 A b \left (\begin {cases} \frac {a^{3} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x} + 2 b \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {\sqrt {x} \log {\left (\sqrt {x} \right )}}{\sqrt {b x}} & \text {otherwise} \end {cases}\right )}{16 b^{2}} + \sqrt {a + b x} \left (- \frac {a^{2} \sqrt {x}}{16 b^{2}} + \frac {a x^{\frac {3}{2}}}{24 b} + \frac {x^{\frac {5}{2}}}{6}\right ) & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{\frac {5}{2}}}{5} & \text {otherwise} \end {cases}\right ) + 2 B a \left (\begin {cases} \frac {a^{3} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x} + 2 b \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {\sqrt {x} \log {\left (\sqrt {x} \right )}}{\sqrt {b x}} & \text {otherwise} \end {cases}\right )}{16 b^{2}} + \sqrt {a + b x} \left (- \frac {a^{2} \sqrt {x}}{16 b^{2}} + \frac {a x^{\frac {3}{2}}}{24 b} + \frac {x^{\frac {5}{2}}}{6}\right ) & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{\frac {5}{2}}}{5} & \text {otherwise} \end {cases}\right ) + 2 B b \left (\begin {cases} - \frac {5 a^{4} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x} + 2 b \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {\sqrt {x} \log {\left (\sqrt {x} \right )}}{\sqrt {b x}} & \text {otherwise} \end {cases}\right )}{128 b^{3}} + \sqrt {a + b x} \left (\frac {5 a^{3} \sqrt {x}}{128 b^{3}} - \frac {5 a^{2} x^{\frac {3}{2}}}{192 b^{2}} + \frac {a x^{\frac {5}{2}}}{48 b} + \frac {x^{\frac {7}{2}}}{8}\right ) & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{\frac {7}{2}}}{7} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (125) = 250\).
Time = 0.21 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.81 \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=\frac {1}{4} \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B x + \frac {1}{2} \, \sqrt {b x^{2} + a x} A a x + \frac {5 \, \sqrt {b x^{2} + a x} B a^{2} x}{32 \, b} - \frac {5 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {5}{2}}} - \frac {A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {3}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} B a^{3}}{64 \, b^{2}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{24 \, b} + \frac {\sqrt {b x^{2} + a x} A a^{2}}{4 \, b} - \frac {\sqrt {b x^{2} + a x} {\left (B a + A b\right )} a x}{4 \, b} + \frac {{\left (B a + A b\right )} a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {5}{2}}} - \frac {\sqrt {b x^{2} + a x} {\left (B a + A b\right )} a^{2}}{8 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (B a + A b\right )}}{3 \, b} \]
[In]
[Out]
Timed out. \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \sqrt {x} (a+b x)^{3/2} (A+B x) \, dx=\int \sqrt {x}\,\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2} \,d x \]
[In]
[Out]