Optimal. Leaf size=102 \[ \frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {684, 654, 634,
212} \begin {gather*} -\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}+\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 634
Rule 654
Rule 684
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {b x+c x^2}} \, dx &=\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {(5 b) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{6 c}\\ &=-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}+\frac {\left (5 b^2\right ) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{8 c^2}\\ &=\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {\left (5 b^3\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^3}\\ &=\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^3}\\ &=\frac {5 b^2 \sqrt {b x+c x^2}}{8 c^3}-\frac {5 b x \sqrt {b x+c x^2}}{12 c^2}+\frac {x^2 \sqrt {b x+c x^2}}{3 c}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 100, normalized size = 0.98 \begin {gather*} \frac {\sqrt {c} x \left (15 b^3+5 b^2 c x-2 b c^2 x^2+8 c^3 x^3\right )+15 b^3 \sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{24 c^{7/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 97, normalized size = 0.95
method | result | size |
risch | \(\frac {\left (8 c^{2} x^{2}-10 b c x +15 b^{2}\right ) x \left (c x +b \right )}{24 c^{3} \sqrt {x \left (c x +b \right )}}-\frac {5 b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {7}{2}}}\) | \(73\) |
default | \(\frac {x^{2} \sqrt {c \,x^{2}+b x}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{6 c}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 88, normalized size = 0.86 \begin {gather*} \frac {\sqrt {c x^{2} + b x} x^{2}}{3 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b x}{12 \, c^{2}} - \frac {5 \, b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.49, size = 147, normalized size = 1.44 \begin {gather*} \left [\frac {15 \, b^{3} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{2} - 10 \, b c^{2} x + 15 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{48 \, c^{4}}, \frac {15 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} x^{2} - 10 \, b c^{2} x + 15 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{24 \, c^{4}}\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}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.96, size = 77, normalized size = 0.75 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, x {\left (\frac {4 \, x}{c} - \frac {5 \, b}{c^{2}}\right )} + \frac {15 \, b^{2}}{c^{3}}\right )} + \frac {5 \, b^{3} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {7}{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}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________