Optimal. Leaf size=69 \[ -\frac {b^2 c}{a \sqrt {a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}}}+\frac {\left (a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}\right )^{3/2}}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2142, 14}
\begin {gather*} \frac {\left (b \sqrt {\frac {a^2 x^2}{b^2}+c}+a x\right )^{3/2}}{3 a}-\frac {b^2 c}{a \sqrt {b \sqrt {\frac {a^2 x^2}{b^2}+c}+a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2142
Rubi steps
\begin {align*} \int \sqrt {a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}} \, dx &=\frac {\text {Subst}\left (\int \frac {b^2 c+x^2}{x^{3/2}} \, dx,x,a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}\right )}{2 a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {b^2 c}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}\right )}{2 a}\\ &=-\frac {b^2 c}{a \sqrt {a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}}}+\frac {\left (a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}\right )^{3/2}}{3 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.53, size = 69, normalized size = 1.00 \begin {gather*} -\frac {b^2 c}{a \sqrt {a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}}}+\frac {\left (a x+b \sqrt {c+\frac {a^2 x^2}{b^2}}\right )^{3/2}}{3 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {a x +b \sqrt {c +\frac {a^{2} x^{2}}{b^{2}}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 59, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (2 \, a x - b \sqrt {\frac {a^{2} x^{2} + b^{2} c}{b^{2}}}\right )} \sqrt {a x + b \sqrt {\frac {a^{2} x^{2} + b^{2} c}{b^{2}}}}}{3 \, a} \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 \sqrt {a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} + c}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \sqrt {a\,x+b\,\sqrt {c+\frac {a^2\,x^2}{b^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________