Optimal. Leaf size=66 \[ \frac {2 x (b+c x) (d x)^m \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-m;\frac {1}{2};\frac {c x}{b}+1\right )}{b \left (b x+c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {674, 67, 65} \[ \frac {2 x (b+c x) (d x)^m \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-m;\frac {1}{2};\frac {c x}{b}+1\right )}{b \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 67
Rule 674
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac {\left (x^{\frac {3}{2}-m} (d x)^m (b+c x)^{3/2}\right ) \int \frac {x^{-\frac {3}{2}+m}}{(b+c x)^{3/2}} \, dx}{\left (b x+c x^2\right )^{3/2}}\\ &=-\frac {\left (c x \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m (b+c x)^{3/2}\right ) \int \frac {\left (-\frac {c x}{b}\right )^{-\frac {3}{2}+m}}{(b+c x)^{3/2}} \, dx}{b \left (b x+c x^2\right )^{3/2}}\\ &=\frac {2 x \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} (d x)^m (b+c x) \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-m;\frac {1}{2};1+\frac {c x}{b}\right )}{b \left (b x+c x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 58, normalized size = 0.88 \[ \frac {2 (d x)^m \left (-\frac {c x}{b}\right )^{\frac {1}{2}-m} \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-m;\frac {1}{2};\frac {c x}{b}+1\right )}{b \sqrt {x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \left (d x\right )^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________