Integrand size = 20, antiderivative size = 95 \[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {a^2 x^2}{b^3 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b c \sqrt {c x^2}}-\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=-\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}}+\frac {a^2 x^2}{b^3 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b c \sqrt {c x^2}} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {x^3}{a+b x} \, dx}{c \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}} \\ & = \frac {a^2 x^2}{b^3 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b c \sqrt {c x^2}}-\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.56 \[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x^3 \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \left (c x^2\right )^{3/2}} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.55
method | result | size |
default | \(-\frac {x^{3} \left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right )}{6 \left (c \,x^{2}\right )^{\frac {3}{2}} b^{4}}\) | \(52\) |
risch | \(\frac {x \left (\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x \right )}{c \sqrt {c \,x^{2}}\, b^{3}}-\frac {a^{3} x \ln \left (b x +a \right )}{b^{4} c \sqrt {c \,x^{2}}}\) | \(63\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.57 \[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{6 \, b^{4} c^{2} x} \]
[In]
[Out]
\[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\int \frac {x^{6}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \]
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.71 \[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x^{4}}{3 \, \sqrt {c x^{2}} b c} - \frac {a x^{3}}{2 \, \sqrt {c x^{2}} b^{2} c} + \frac {a^{2} x^{2}}{\sqrt {c x^{2}} b^{3} c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4} c^{\frac {3}{2}}} + \frac {29 \, a^{3} x}{6 \, \sqrt {c x^{2}} b^{4} c} - \frac {a^{3} \log \left (b x\right )}{b^{4} c^{\frac {3}{2}}} - \frac {2 \, a^{4}}{\sqrt {c x^{2}} b^{5} c} + \frac {2 \, a^{4}}{b^{5} c^{\frac {3}{2}} x} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {\frac {6 \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{4} \sqrt {c}} - \frac {6 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} \sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {2 \, b^{2} c x^{3} - 3 \, a b c x^{2} + 6 \, a^{2} c x}{b^{3} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}}{6 \, c} \]
[In]
[Out]
Timed out. \[ \int \frac {x^6}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\int \frac {x^6}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \]
[In]
[Out]