3.670 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{5/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{2 a^2}{3 d (d x)^{3/2}}+\frac{4 a b \sqrt{d x}}{d^3}+\frac{2 b^2 (d x)^{5/2}}{5 d^5} \]

[Out]

(-2*a^2)/(3*d*(d*x)^(3/2)) + (4*a*b*Sqrt[d*x])/d^3 + (2*b^2*(d*x)^(5/2))/(5*d^5)

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Rubi [A]  time = 0.0144974, 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} \[ -\frac{2 a^2}{3 d (d x)^{3/2}}+\frac{4 a b \sqrt{d x}}{d^3}+\frac{2 b^2 (d x)^{5/2}}{5 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(5/2),x]

[Out]

(-2*a^2)/(3*d*(d*x)^(3/2)) + (4*a*b*Sqrt[d*x])/d^3 + (2*b^2*(d*x)^(5/2))/(5*d^5)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{5/2}} \, dx &=\int \left (\frac{a^2}{(d x)^{5/2}}+\frac{2 a b}{d^2 \sqrt{d x}}+\frac{b^2 (d x)^{3/2}}{d^4}\right ) \, dx\\ &=-\frac{2 a^2}{3 d (d x)^{3/2}}+\frac{4 a b \sqrt{d x}}{d^3}+\frac{2 b^2 (d x)^{5/2}}{5 d^5}\\ \end{align*}

Mathematica [A]  time = 0.011717, size = 33, normalized size = 0.67 \[ \frac{x \left (-10 a^2+60 a b x^2+6 b^2 x^4\right )}{15 (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(5/2),x]

[Out]

(x*(-10*a^2 + 60*a*b*x^2 + 6*b^2*x^4))/(15*(d*x)^(5/2))

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Maple [A]  time = 0.046, size = 30, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -6\,{b}^{2}{x}^{4}-60\,ab{x}^{2}+10\,{a}^{2} \right ) x}{15} \left ( dx \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(5/2),x)

[Out]

-2/15*(-3*b^2*x^4-30*a*b*x^2+5*a^2)*x/(d*x)^(5/2)

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Maxima [A]  time = 0.975606, size = 58, normalized size = 1.18 \begin{align*} -\frac{2 \,{\left (\frac{5 \, a^{2}}{\left (d x\right )^{\frac{3}{2}}} - \frac{3 \,{\left (\left (d x\right )^{\frac{5}{2}} b^{2} + 10 \, \sqrt{d x} a b d^{2}\right )}}{d^{4}}\right )}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(5/2),x, algorithm="maxima")

[Out]

-2/15*(5*a^2/(d*x)^(3/2) - 3*((d*x)^(5/2)*b^2 + 10*sqrt(d*x)*a*b*d^2)/d^4)/d

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Fricas [A]  time = 1.30104, size = 80, normalized size = 1.63 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{4} + 30 \, a b x^{2} - 5 \, a^{2}\right )} \sqrt{d x}}{15 \, d^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*b^2*x^4 + 30*a*b*x^2 - 5*a^2)*sqrt(d*x)/(d^3*x^2)

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Sympy [A]  time = 0.880359, size = 48, normalized size = 0.98 \begin{align*} - \frac{2 a^{2}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} + \frac{4 a b \sqrt{x}}{d^{\frac{5}{2}}} + \frac{2 b^{2} x^{\frac{5}{2}}}{5 d^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)/(d*x)**(5/2),x)

[Out]

-2*a**2/(3*d**(5/2)*x**(3/2)) + 4*a*b*sqrt(x)/d**(5/2) + 2*b**2*x**(5/2)/(5*d**(5/2))

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Giac [A]  time = 1.12469, size = 72, normalized size = 1.47 \begin{align*} -\frac{2 \,{\left (\frac{5 \, a^{2} d}{\sqrt{d x} x} - \frac{3 \,{\left (\sqrt{d x} b^{2} d^{10} x^{2} + 10 \, \sqrt{d x} a b d^{10}\right )}}{d^{10}}\right )}}{15 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(5/2),x, algorithm="giac")

[Out]

-2/15*(5*a^2*d/(sqrt(d*x)*x) - 3*(sqrt(d*x)*b^2*d^10*x^2 + 10*sqrt(d*x)*a*b*d^10)/d^10)/d^3