Optimal. Leaf size=49 \[ -\frac{2 a^2}{5 d (d x)^{5/2}}-\frac{4 a b}{d^3 \sqrt{d x}}+\frac{2 b^2 (d x)^{3/2}}{3 d^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0130246, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {14} \[ -\frac{2 a^2}{5 d (d x)^{5/2}}-\frac{4 a b}{d^3 \sqrt{d x}}+\frac{2 b^2 (d x)^{3/2}}{3 d^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rubi steps
\begin{align*} \int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{7/2}} \, dx &=\int \left (\frac{a^2}{(d x)^{7/2}}+\frac{2 a b}{d^2 (d x)^{3/2}}+\frac{b^2 \sqrt{d x}}{d^4}\right ) \, dx\\ &=-\frac{2 a^2}{5 d (d x)^{5/2}}-\frac{4 a b}{d^3 \sqrt{d x}}+\frac{2 b^2 (d x)^{3/2}}{3 d^5}\\ \end{align*}
Mathematica [A] time = 0.0127884, size = 38, normalized size = 0.78 \[ \frac{2 \sqrt{d x} \left (-3 a^2-30 a b x^2+5 b^2 x^4\right )}{15 d^4 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 30, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -10\,{b}^{2}{x}^{4}+60\,ab{x}^{2}+6\,{a}^{2} \right ) x}{15} \left ( dx \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.968787, size = 63, normalized size = 1.29 \begin{align*} \frac{2 \,{\left (\frac{5 \, \left (d x\right )^{\frac{3}{2}} b^{2}}{d^{4}} - \frac{3 \,{\left (10 \, a b d^{2} x^{2} + a^{2} d^{2}\right )}}{\left (d x\right )^{\frac{5}{2}} d^{2}}\right )}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.2439, size = 80, normalized size = 1.63 \begin{align*} \frac{2 \,{\left (5 \, b^{2} x^{4} - 30 \, a b x^{2} - 3 \, a^{2}\right )} \sqrt{d x}}{15 \, d^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.80933, size = 48, normalized size = 0.98 \begin{align*} - \frac{2 a^{2}}{5 d^{\frac{7}{2}} x^{\frac{5}{2}}} - \frac{4 a b}{d^{\frac{7}{2}} \sqrt{x}} + \frac{2 b^{2} x^{\frac{3}{2}}}{3 d^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13409, size = 65, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (5 \, \sqrt{d x} b^{2} x - \frac{3 \,{\left (10 \, a b d^{3} x^{2} + a^{2} d^{3}\right )}}{\sqrt{d x} d^{2} x^{2}}\right )}}{15 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]