Integrand size = 22, antiderivative size = 119 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} \sqrt {c}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} \sqrt {c}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 a^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2} \]
[In]
[Out]
Rule 95
Rule 96
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {(3 (b c-a d)) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}} \, dx}{4 a} \\ & = \frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}+\frac {\left (3 (b c-a d)^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^2} \\ & = \frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}+\frac {\left (3 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^2} \\ & = \frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} \sqrt {c}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} (2 a c-3 b c x+5 a d x)}{4 a^2 x^2}-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} \sqrt {c}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(93)=186\).
Time = 1.65 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.14
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2}+10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right )}{8 a^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {a c}}\) | \(255\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.81 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - {\left (3 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{3} c x^{2}}, \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - {\left (3 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{3} c x^{2}}\right ] \]
[In]
[Out]
\[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{3} \sqrt {a + b x}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1078 vs. \(2 (93) = 186\).
Time = 0.69 (sec) , antiderivative size = 1078, normalized size of antiderivative = 9.06 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=-\frac {\frac {3 \, {\left (\sqrt {b d} b^{3} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c d {\left | b \right |} + \sqrt {b d} a^{2} b d^{2} {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{2} b} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{9} c^{5} {\left | b \right |} - 17 \, \sqrt {b d} a b^{8} c^{4} d {\left | b \right |} + 38 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} {\left | b \right |} - 42 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} {\left | b \right |} + 23 \, \sqrt {b d} a^{4} b^{5} c d^{4} {\left | b \right |} - 5 \, \sqrt {b d} a^{5} b^{4} d^{5} {\left | b \right |} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{7} c^{4} {\left | b \right |} + 20 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{6} c^{3} d {\left | b \right |} + 2 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} c^{2} d^{2} {\left | b \right |} - 28 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{4} c d^{3} {\left | b \right |} + 15 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{3} d^{4} {\left | b \right |} + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{5} c^{3} {\left | b \right |} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{4} c^{2} d {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{3} c d^{2} {\left | b \right |} - 15 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{2} d^{3} {\left | b \right |} - 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{3} c^{2} {\left | b \right |} + 6 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{2} c d {\left | b \right |} + 5 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b d^{2} {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2} a^{2}}}{4 \, b} \]
[In]
[Out]
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^3\,\sqrt {a+b\,x}} \,d x \]
[In]
[Out]