3.671 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{7/2}} \, dx\)

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]

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

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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]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(7/2),x]

[Out]

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

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(7/2),x]

[Out]

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

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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]

int((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(7/2),x)

[Out]

-2/15*(-5*b^2*x^4+30*a*b*x^2+3*a^2)*x/(d*x)^(7/2)

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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]

integrate((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(7/2),x, algorithm="maxima")

[Out]

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

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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]

integrate((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(7/2),x, algorithm="fricas")

[Out]

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

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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]

integrate((b**2*x**4+2*a*b*x**2+a**2)/(d*x)**(7/2),x)

[Out]

-2*a**2/(5*d**(7/2)*x**(5/2)) - 4*a*b/(d**(7/2)*sqrt(x)) + 2*b**2*x**(3/2)/(3*d**(7/2))

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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]

integrate((b^2*x^4+2*a*b*x^2+a^2)/(d*x)^(7/2),x, algorithm="giac")

[Out]

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