Optimal. Leaf size=195 \[ \frac {6 a b^2 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left (a+b x^2\right )}+\frac {2 a^3 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 195, 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} \begin {gather*} \frac {2 b^3 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left (a+b x^2\right )}+\frac {6 a b^2 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac {2 a^3 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 1112
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \sqrt {d x} \left (a b+b^2 x^2\right )^3 \, 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 (a^3 b^3 \sqrt {d x}+\frac {3 a^2 b^4 (d x)^{5/2}}{d^2}+\frac {3 a b^5 (d x)^{9/2}}{d^4}+\frac {b^6 (d x)^{13/2}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {2 a^3 (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac {6 a b^2 (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left (a+b x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 66, normalized size = 0.34 \begin {gather*} \frac {2 \sqrt {d x} \sqrt {\left (a+b x^2\right )^2} \left (385 a^3 x+495 a^2 b x^3+315 a b^2 x^5+77 b^3 x^7\right )}{1155 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 83.78, size = 96, normalized size = 0.49 \begin {gather*} \frac {2 (d x)^{3/2} \left (a d^2+b d^2 x^2\right ) \left (385 a^3 d^6+495 a^2 b d^6 x^2+315 a b^2 d^6 x^4+77 b^3 d^6 x^6\right )}{1155 d^9 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.77, size = 40, normalized size = 0.21 \begin {gather*} \frac {2}{1155} \, {\left (77 \, b^{3} x^{7} + 315 \, a b^{2} x^{5} + 495 \, a^{2} b x^{3} + 385 \, a^{3} x\right )} \sqrt {d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 85, normalized size = 0.44 \begin {gather*} \frac {2}{15} \, \sqrt {d x} b^{3} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {6}{11} \, \sqrt {d x} a b^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {6}{7} \, \sqrt {d x} a^{2} b x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{3} \, \sqrt {d x} a^{3} x \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 61, normalized size = 0.31 \begin {gather*} \frac {2 \left (77 b^{3} x^{6}+315 a \,b^{2} x^{4}+495 a^{2} b \,x^{2}+385 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} \sqrt {d x}\, x}{1155 \left (b \,x^{2}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.44, size = 83, normalized size = 0.43 \begin {gather*} \frac {2}{165} \, {\left (11 \, b^{3} \sqrt {d} x^{3} + 15 \, a b^{2} \sqrt {d} x\right )} x^{\frac {9}{2}} + \frac {4}{77} \, {\left (7 \, a b^{2} \sqrt {d} x^{3} + 11 \, a^{2} b \sqrt {d} x\right )} x^{\frac {5}{2}} + \frac {2}{21} \, {\left (3 \, a^{2} b \sqrt {d} x^{3} + 7 \, a^{3} \sqrt {d} x\right )} \sqrt {x} \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 {d\,x}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \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 {d x} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________