Optimal. Leaf size=48 \[ \frac {4 b \sqrt {x}}{c^2 \sqrt {b x+c x^2}}+\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662}
\begin {gather*} \frac {4 b \sqrt {x}}{c^2 \sqrt {b x+c x^2}}+\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 662
Rule 670
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}}-\frac {(2 b) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac {4 b \sqrt {x}}{c^2 \sqrt {b x+c x^2}}+\frac {2 x^{3/2}}{c \sqrt {b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 28, normalized size = 0.58 \begin {gather*} \frac {2 \sqrt {x} (2 b+c x)}{c^2 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.44, size = 32, normalized size = 0.67
method | result | size |
gosper | \(\frac {2 \left (c x +b \right ) \left (c x +2 b \right ) x^{\frac {3}{2}}}{c^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\) | \(32\) |
default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \left (c x +2 b \right )}{\sqrt {x}\, \left (c x +b \right ) c^{2}}\) | \(32\) |
risch | \(\frac {2 \left (c x +b \right ) \sqrt {x}}{c^{2} \sqrt {x \left (c x +b \right )}}+\frac {2 b \sqrt {x}}{c^{2} \sqrt {x \left (c x +b \right )}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.71, size = 39, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (c x + 2 \, b\right )} \sqrt {x}}{c^{3} x^{2} + b c^{2} x} \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^{\frac {5}{2}}}{\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 = 2.16, size = 38, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {c x + b}}{c} + \frac {b}{\sqrt {c x + b} c}\right )}}{c} - \frac {4 \, \sqrt {b}}{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 = -0.02 \begin {gather*} \int \frac {x^{5/2}}{{\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]
________________________________________________________________________________________