Optimal. Leaf size=68 \[ -\frac {d^2 x (d x)^{m-2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m-2,-n;m-1;-\frac {b x}{a}\right )}{c (2-m) \sqrt {c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 16, 66, 64} \[ -\frac {d^2 x (d x)^{m-2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m-2,-n;m-1;-\frac {b x}{a}\right )}{c (2-m) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 16
Rule 64
Rule 66
Rubi steps
\begin {align*} \int \frac {(d x)^m (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(d x)^m (a+b x)^n}{x^3} \, dx}{c \sqrt {c x^2}}\\ &=\frac {\left (d^3 x\right ) \int (d x)^{-3+m} (a+b x)^n \, dx}{c \sqrt {c x^2}}\\ &=\frac {\left (d^3 x (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int (d x)^{-3+m} \left (1+\frac {b x}{a}\right )^n \, dx}{c \sqrt {c x^2}}\\ &=-\frac {d^2 x (d x)^{-2+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-2+m,-n;-1+m;-\frac {b x}{a}\right )}{c (2-m) \sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 57, normalized size = 0.84 \[ \frac {x (d x)^m (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m-2,-n;m-1;-\frac {b x}{a}\right )}{(m-2) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2}} {\left (b x + a\right )}^{n} \left (d x\right )^{m}}{c^{2} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m} \left (b x +a \right )^{n}}{\left (c \,x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,x\right )}^m\,{\left (a+b\,x\right )}^n}{{\left (c\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m} \left (a + b x\right )^{n}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________