3.1052 \(\int \frac{(a+b x^2+c x^4)^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 a^2}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left (2 a c+b^2\right )+\frac{4}{3} a b x^{3/2}+\frac{4}{11} b c x^{11/2}+\frac{2}{15} c^2 x^{15/2} \]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*(b^2 + 2*a*c)*x^(7/2))/7 + (4*b*c*x^(11/2))/11 + (2*c^2*x^(15/2))/15

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Rubi [A]  time = 0.0222581, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1108} \[ -\frac{2 a^2}{\sqrt{x}}+\frac{2}{7} x^{7/2} \left (2 a c+b^2\right )+\frac{4}{3} a b x^{3/2}+\frac{4}{11} b c x^{11/2}+\frac{2}{15} c^2 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*(b^2 + 2*a*c)*x^(7/2))/7 + (4*b*c*x^(11/2))/11 + (2*c^2*x^(15/2))/15

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

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

Mathematica [A]  time = 0.04392, size = 54, normalized size = 0.87 \[ \frac{2 \left (-1155 a^2+110 a \left (7 b x^2+3 c x^4\right )+165 b^2 x^4+210 b c x^6+77 c^2 x^8\right )}{1155 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1155*a^2 + 165*b^2*x^4 + 210*b*c*x^6 + 77*c^2*x^8 + 110*a*(7*b*x^2 + 3*c*x^4)))/(1155*Sqrt[x])

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Maple [A]  time = 0.048, size = 49, normalized size = 0.8 \begin{align*} -{\frac{-154\,{c}^{2}{x}^{8}-420\,bc{x}^{6}-660\,{x}^{4}ac-330\,{b}^{2}{x}^{4}-1540\,ab{x}^{2}+2310\,{a}^{2}}{1155}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^(3/2),x)

[Out]

-2/1155*(-77*c^2*x^8-210*b*c*x^6-330*a*c*x^4-165*b^2*x^4-770*a*b*x^2+1155*a^2)/x^(1/2)

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Maxima [A]  time = 0.964376, size = 59, normalized size = 0.95 \begin{align*} \frac{2}{15} \, c^{2} x^{\frac{15}{2}} + \frac{4}{11} \, b c x^{\frac{11}{2}} + \frac{2}{7} \,{\left (b^{2} + 2 \, a c\right )} x^{\frac{7}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^(3/2),x, algorithm="maxima")

[Out]

2/15*c^2*x^(15/2) + 4/11*b*c*x^(11/2) + 2/7*(b^2 + 2*a*c)*x^(7/2) + 4/3*a*b*x^(3/2) - 2*a^2/sqrt(x)

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Fricas [A]  time = 1.28154, size = 124, normalized size = 2. \begin{align*} \frac{2 \,{\left (77 \, c^{2} x^{8} + 210 \, b c x^{6} + 165 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 770 \, a b x^{2} - 1155 \, a^{2}\right )}}{1155 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^(3/2),x, algorithm="fricas")

[Out]

2/1155*(77*c^2*x^8 + 210*b*c*x^6 + 165*(b^2 + 2*a*c)*x^4 + 770*a*b*x^2 - 1155*a^2)/sqrt(x)

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Sympy [A]  time = 5.53825, size = 68, normalized size = 1.1 \begin{align*} - \frac{2 a^{2}}{\sqrt{x}} + \frac{4 a b x^{\frac{3}{2}}}{3} + \frac{4 a c x^{\frac{7}{2}}}{7} + \frac{2 b^{2} x^{\frac{7}{2}}}{7} + \frac{4 b c x^{\frac{11}{2}}}{11} + \frac{2 c^{2} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**(3/2),x)

[Out]

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

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Giac [A]  time = 1.16571, size = 62, normalized size = 1. \begin{align*} \frac{2}{15} \, c^{2} x^{\frac{15}{2}} + \frac{4}{11} \, b c x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, a c x^{\frac{7}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^(3/2),x, algorithm="giac")

[Out]

2/15*c^2*x^(15/2) + 4/11*b*c*x^(11/2) + 2/7*b^2*x^(7/2) + 4/7*a*c*x^(7/2) + 4/3*a*b*x^(3/2) - 2*a^2/sqrt(x)