3.669 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{(d x)^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{2 a^2}{d \sqrt{d x}}+\frac{4 a b (d x)^{3/2}}{3 d^3}+\frac{2 b^2 (d x)^{7/2}}{7 d^5} \]

[Out]

(-2*a^2)/(d*Sqrt[d*x]) + (4*a*b*(d*x)^(3/2))/(3*d^3) + (2*b^2*(d*x)^(7/2))/(7*d^5)

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Rubi [A]  time = 0.0149061, 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} \[ -\frac{2 a^2}{d \sqrt{d x}}+\frac{4 a b (d x)^{3/2}}{3 d^3}+\frac{2 b^2 (d x)^{7/2}}{7 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(3/2),x]

[Out]

(-2*a^2)/(d*Sqrt[d*x]) + (4*a*b*(d*x)^(3/2))/(3*d^3) + (2*b^2*(d*x)^(7/2))/(7*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)^{3/2}} \, dx &=\int \left (\frac{a^2}{(d x)^{3/2}}+\frac{2 a b \sqrt{d x}}{d^2}+\frac{b^2 (d x)^{5/2}}{d^4}\right ) \, dx\\ &=-\frac{2 a^2}{d \sqrt{d x}}+\frac{4 a b (d x)^{3/2}}{3 d^3}+\frac{2 b^2 (d x)^{7/2}}{7 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0114596, size = 33, normalized size = 0.67 \[ \frac{2 x \left (-21 a^2+14 a b x^2+3 b^2 x^4\right )}{21 (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/(d*x)^(3/2),x]

[Out]

(2*x*(-21*a^2 + 14*a*b*x^2 + 3*b^2*x^4))/(21*(d*x)^(3/2))

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Maple [A]  time = 0.049, size = 30, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -6\,{b}^{2}{x}^{4}-28\,ab{x}^{2}+42\,{a}^{2} \right ) x}{21} \left ( dx \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

-2/21*(-3*b^2*x^4-14*a*b*x^2+21*a^2)*x/(d*x)^(3/2)

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Maxima [A]  time = 0.972855, size = 59, normalized size = 1.2 \begin{align*} -\frac{2 \,{\left (\frac{21 \, a^{2}}{\sqrt{d x}} - \frac{3 \, \left (d x\right )^{\frac{7}{2}} b^{2} + 14 \, \left (d x\right )^{\frac{3}{2}} a b d^{2}}{d^{4}}\right )}}{21 \, 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)^(3/2),x, algorithm="maxima")

[Out]

-2/21*(21*a^2/sqrt(d*x) - (3*(d*x)^(7/2)*b^2 + 14*(d*x)^(3/2)*a*b*d^2)/d^4)/d

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Fricas [A]  time = 1.1699, size = 78, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{4} + 14 \, a b x^{2} - 21 \, a^{2}\right )} \sqrt{d x}}{21 \, d^{2} x} \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)^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*b^2*x^4 + 14*a*b*x^2 - 21*a^2)*sqrt(d*x)/(d^2*x)

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Sympy [A]  time = 0.623172, size = 48, normalized size = 0.98 \begin{align*} - \frac{2 a^{2}}{d^{\frac{3}{2}} \sqrt{x}} + \frac{4 a b x^{\frac{3}{2}}}{3 d^{\frac{3}{2}}} + \frac{2 b^{2} x^{\frac{7}{2}}}{7 d^{\frac{3}{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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.107, size = 69, normalized size = 1.41 \begin{align*} -\frac{2 \,{\left (\frac{21 \, a^{2}}{\sqrt{d x}} - \frac{3 \, \sqrt{d x} b^{2} d^{27} x^{3} + 14 \, \sqrt{d x} a b d^{27} x}{d^{28}}\right )}}{21 \, 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)^(3/2),x, algorithm="giac")

[Out]

-2/21*(21*a^2/sqrt(d*x) - (3*sqrt(d*x)*b^2*d^27*x^3 + 14*sqrt(d*x)*a*b*d^27*x)/d^28)/d