Optimal. Leaf size=88 \[ \frac {2 (d x)^{m+1} \left (a+b \sqrt {c x^2}\right )^{3/2} \left (-\frac {b \sqrt {c x^2}}{a}\right )^{-m} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};\frac {\sqrt {c x^2} b}{a}+1\right )}{3 b d \sqrt {c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {368, 67, 65} \[ \frac {2 (d x)^{m+1} \left (a+b \sqrt {c x^2}\right )^{3/2} \left (-\frac {b \sqrt {c x^2}}{a}\right )^{-m} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};\frac {\sqrt {c x^2} b}{a}+1\right )}{3 b d \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 67
Rule 368
Rubi steps
\begin {align*} \int (d x)^m \sqrt {a+b \sqrt {c x^2}} \, dx &=\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)}\right ) \operatorname {Subst}\left (\int x^m \sqrt {a+b x} \, dx,x,\sqrt {c x^2}\right )}{d}\\ &=\frac {\left ((d x)^{1+m} \left (c x^2\right )^{\frac {1}{2} (-1-m)+\frac {m}{2}} \left (-\frac {b \sqrt {c x^2}}{a}\right )^{-m}\right ) \operatorname {Subst}\left (\int \left (-\frac {b x}{a}\right )^m \sqrt {a+b x} \, dx,x,\sqrt {c x^2}\right )}{d}\\ &=\frac {2 (d x)^{1+m} \left (-\frac {b \sqrt {c x^2}}{a}\right )^{-m} \left (a+b \sqrt {c x^2}\right )^{3/2} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};1+\frac {b \sqrt {c x^2}}{a}\right )}{3 b d \sqrt {c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 74, normalized size = 0.84 \[ \frac {x (d x)^m \sqrt {a+b \sqrt {c x^2}} \, _2F_1\left (-\frac {1}{2},m+1;m+2;-\frac {b \sqrt {c x^2}}{a}\right )}{(m+1) \sqrt {\frac {b \sqrt {c x^2}}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {\sqrt {c x^{2}} b + a} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {c x^{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\sqrt {c \,x^{2}}\, b}\, \left (d x \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {c x^{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,\sqrt {a+b\,\sqrt {c\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \sqrt {a + b \sqrt {c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________