Optimal. Leaf size=193 \[ \frac {2 a b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {2 a^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1112, 270} \[ \frac {2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac {2 a b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac {2 a^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 1112
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\sqrt {d x}} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^3}{\sqrt {d x}} \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a^3 b^3}{\sqrt {d x}}+\frac {3 a^2 b^4 (d x)^{3/2}}{d^2}+\frac {3 a b^5 (d x)^{7/2}}{d^4}+\frac {b^6 (d x)^{11/2}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {2 a^3 \sqrt {d x} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac {2 a b^2 (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{13/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 66, normalized size = 0.34 \[ \frac {2 \sqrt {\left (a+b x^2\right )^2} \left (195 a^3 x+117 a^2 b x^3+65 a b^2 x^5+15 b^3 x^7\right )}{195 \sqrt {d x} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.94, size = 42, normalized size = 0.22 \[ \frac {2 \, {\left (15 \, b^{3} x^{6} + 65 \, a b^{2} x^{4} + 117 \, a^{2} b x^{2} + 195 \, a^{3}\right )} \sqrt {d x}}{195 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 89, normalized size = 0.46 \[ \frac {2 \, {\left (15 \, \sqrt {d x} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 65 \, \sqrt {d x} a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 117 \, \sqrt {d x} a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 195 \, \sqrt {d x} a^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{195 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 61, normalized size = 0.32 \[ \frac {2 \left (15 b^{3} x^{6}+65 a \,b^{2} x^{4}+117 a^{2} b \,x^{2}+195 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x}{195 \left (b \,x^{2}+a \right )^{3} \sqrt {d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.46, size = 87, normalized size = 0.45 \[ \frac {2 \, {\left (5 \, {\left (9 \, b^{3} \sqrt {d} x^{3} + 13 \, a b^{2} \sqrt {d} x\right )} x^{\frac {7}{2}} + 26 \, {\left (5 \, a b^{2} \sqrt {d} x^{3} + 9 \, a^{2} b \sqrt {d} x\right )} x^{\frac {3}{2}} + \frac {117 \, {\left (a^{2} b \sqrt {d} x^{3} + 5 \, a^{3} \sqrt {d} x\right )}}{\sqrt {x}}\right )}}{585 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.50, size = 76, normalized size = 0.39 \[ \frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (\frac {6\,a^2\,x^3}{5}+\frac {2\,b^2\,x^7}{13}+\frac {2\,a^3\,x}{b}+\frac {2\,a\,b\,x^5}{3}\right )}{x^2\,\sqrt {d\,x}+\frac {a\,\sqrt {d\,x}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{\sqrt {d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________