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

Optimal. Leaf size=89 \[ \frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}-\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]

[Out]

(-2*a^4)/(d*Sqrt[d*x]) + (8*a^3*b*(d*x)^(3/2))/(3*d^3) + (12*a^2*b^2*(d*x)^(7/2))/(7*d^5) + (8*a*b^3*(d*x)^(11
/2))/(11*d^7) + (2*b^4*(d*x)^(15/2))/(15*d^9)

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Rubi [A]  time = 0.0420059, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {28, 270} \[ \frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}-\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^4)/(d*Sqrt[d*x]) + (8*a^3*b*(d*x)^(3/2))/(3*d^3) + (12*a^2*b^2*(d*x)^(7/2))/(7*d^5) + (8*a*b^3*(d*x)^(11
/2))/(11*d^7) + (2*b^4*(d*x)^(15/2))/(15*d^9)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{(d x)^{3/2}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{(d x)^{3/2}} \, dx}{b^4}\\ &=\frac{\int \left (\frac{a^4 b^4}{(d x)^{3/2}}+\frac{4 a^3 b^5 \sqrt{d x}}{d^2}+\frac{6 a^2 b^6 (d x)^{5/2}}{d^4}+\frac{4 a b^7 (d x)^{9/2}}{d^6}+\frac{b^8 (d x)^{13/2}}{d^8}\right ) \, dx}{b^4}\\ &=-\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}+\frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9}\\ \end{align*}

Mathematica [A]  time = 0.0160557, size = 55, normalized size = 0.62 \[ \frac{2 x \left (990 a^2 b^2 x^4+1540 a^3 b x^2-1155 a^4+420 a b^3 x^6+77 b^4 x^8\right )}{1155 (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-1155*a^4 + 1540*a^3*b*x^2 + 990*a^2*b^2*x^4 + 420*a*b^3*x^6 + 77*b^4*x^8))/(1155*(d*x)^(3/2))

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Maple [A]  time = 0.049, size = 52, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -154\,{b}^{4}{x}^{8}-840\,a{b}^{3}{x}^{6}-1980\,{a}^{2}{b}^{2}{x}^{4}-3080\,{a}^{3}b{x}^{2}+2310\,{a}^{4} \right ) x}{1155} \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)^2/(d*x)^(3/2),x)

[Out]

-2/1155*(-77*b^4*x^8-420*a*b^3*x^6-990*a^2*b^2*x^4-1540*a^3*b*x^2+1155*a^4)*x/(d*x)^(3/2)

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Maxima [A]  time = 0.983885, size = 103, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \left (d x\right )^{\frac{15}{2}} b^{4} + 420 \, \left (d x\right )^{\frac{11}{2}} a b^{3} d^{2} + 990 \, \left (d x\right )^{\frac{7}{2}} a^{2} b^{2} d^{4} + 1540 \, \left (d x\right )^{\frac{3}{2}} a^{3} b d^{6}}{d^{8}}\right )}}{1155 \, 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)^2/(d*x)^(3/2),x, algorithm="maxima")

[Out]

-2/1155*(1155*a^4/sqrt(d*x) - (77*(d*x)^(15/2)*b^4 + 420*(d*x)^(11/2)*a*b^3*d^2 + 990*(d*x)^(7/2)*a^2*b^2*d^4
+ 1540*(d*x)^(3/2)*a^3*b*d^6)/d^8)/d

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Fricas [A]  time = 1.19973, size = 136, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (77 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 990 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} - 1155 \, a^{4}\right )} \sqrt{d x}}{1155 \, 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)^2/(d*x)^(3/2),x, algorithm="fricas")

[Out]

2/1155*(77*b^4*x^8 + 420*a*b^3*x^6 + 990*a^2*b^2*x^4 + 1540*a^3*b*x^2 - 1155*a^4)*sqrt(d*x)/(d^2*x)

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Sympy [A]  time = 1.3449, size = 88, normalized size = 0.99 \begin{align*} - \frac{2 a^{4}}{d^{\frac{3}{2}} \sqrt{x}} + \frac{8 a^{3} b x^{\frac{3}{2}}}{3 d^{\frac{3}{2}}} + \frac{12 a^{2} b^{2} x^{\frac{7}{2}}}{7 d^{\frac{3}{2}}} + \frac{8 a b^{3} x^{\frac{11}{2}}}{11 d^{\frac{3}{2}}} + \frac{2 b^{4} x^{\frac{15}{2}}}{15 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)**2/(d*x)**(3/2),x)

[Out]

-2*a**4/(d**(3/2)*sqrt(x)) + 8*a**3*b*x**(3/2)/(3*d**(3/2)) + 12*a**2*b**2*x**(7/2)/(7*d**(3/2)) + 8*a*b**3*x*
*(11/2)/(11*d**(3/2)) + 2*b**4*x**(15/2)/(15*d**(3/2))

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Giac [A]  time = 1.12669, size = 120, normalized size = 1.35 \begin{align*} -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \sqrt{d x} b^{4} d^{119} x^{7} + 420 \, \sqrt{d x} a b^{3} d^{119} x^{5} + 990 \, \sqrt{d x} a^{2} b^{2} d^{119} x^{3} + 1540 \, \sqrt{d x} a^{3} b d^{119} x}{d^{120}}\right )}}{1155 \, 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)^2/(d*x)^(3/2),x, algorithm="giac")

[Out]

-2/1155*(1155*a^4/sqrt(d*x) - (77*sqrt(d*x)*b^4*d^119*x^7 + 420*sqrt(d*x)*a*b^3*d^119*x^5 + 990*sqrt(d*x)*a^2*
b^2*d^119*x^3 + 1540*sqrt(d*x)*a^3*b*d^119*x)/d^120)/d