Optimal. Leaf size=69 \[ -\frac {2 x^2}{c \sqrt {b x+c x^2}}+\frac {3 \sqrt {b x+c x^2}}{c^2}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {682, 654, 634,
212} \begin {gather*} -\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}}+\frac {3 \sqrt {b x+c x^2}}{c^2}-\frac {2 x^2}{c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 634
Rule 654
Rule 682
Rubi steps
\begin {align*} \int \frac {x^3}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 x^2}{c \sqrt {b x+c x^2}}+\frac {3 \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{c}\\ &=-\frac {2 x^2}{c \sqrt {b x+c x^2}}+\frac {3 \sqrt {b x+c x^2}}{c^2}-\frac {(3 b) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c^2}\\ &=-\frac {2 x^2}{c \sqrt {b x+c x^2}}+\frac {3 \sqrt {b x+c x^2}}{c^2}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^2}\\ &=-\frac {2 x^2}{c \sqrt {b x+c x^2}}+\frac {3 \sqrt {b x+c x^2}}{c^2}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 72, normalized size = 1.04 \begin {gather*} \frac {\sqrt {c} x (3 b+c x)+3 b \sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{c^{5/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.46, size = 119, normalized size = 1.72
method | result | size |
risch | \(\frac {x \left (c x +b \right )}{c^{2} \sqrt {x \left (c x +b \right )}}-\frac {3 b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {5}{2}}}+\frac {2 b \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{c^{3} \left (\frac {b}{c}+x \right )}\) | \(90\) |
default | \(\frac {x^{2}}{c \sqrt {c \,x^{2}+b x}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 66, normalized size = 0.96 \begin {gather*} \frac {x^{2}}{\sqrt {c x^{2} + b x} c} + \frac {3 \, b x}{\sqrt {c x^{2} + b x} c^{2}} - \frac {3 \, b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.73, size = 152, normalized size = 2.20 \begin {gather*} \left [\frac {3 \, {\left (b c x + b^{2}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (c^{2} x + 3 \, b c\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (c^{4} x + b c^{3}\right )}}, \frac {3 \, {\left (b c x + b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (c^{2} x + 3 \, b c\right )} \sqrt {c x^{2} + b x}}{c^{4} x + b c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.65, size = 89, normalized size = 1.29 \begin {gather*} \frac {3 \, b \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {5}{2}}} + \frac {2 \, b^{2}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} c + b \sqrt {c}\right )} c^{2}} + \frac {\sqrt {c x^{2} + b x}}{c^{2}} \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 \frac {x^3}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________