3.1054 \(\int \frac{(a+b x^2+c x^4)^2}{x^{7/2}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 a^2}{5 x^{5/2}}+\frac{2}{3} x^{3/2} \left (2 a c+b^2\right )-\frac{4 a b}{\sqrt{x}}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2} \]

[Out]

(-2*a^2)/(5*x^(5/2)) - (4*a*b)/Sqrt[x] + (2*(b^2 + 2*a*c)*x^(3/2))/3 + (4*b*c*x^(7/2))/7 + (2*c^2*x^(11/2))/11

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Rubi [A]  time = 0.0229676, 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} \[ -\frac{2 a^2}{5 x^{5/2}}+\frac{2}{3} x^{3/2} \left (2 a c+b^2\right )-\frac{4 a b}{\sqrt{x}}+\frac{4}{7} b c x^{7/2}+\frac{2}{11} c^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^(7/2),x]

[Out]

(-2*a^2)/(5*x^(5/2)) - (4*a*b)/Sqrt[x] + (2*(b^2 + 2*a*c)*x^(3/2))/3 + (4*b*c*x^(7/2))/7 + (2*c^2*x^(11/2))/11

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

Mathematica [A]  time = 0.0445557, size = 50, normalized size = 0.81 \[ \frac{2 \left (-231 a^2+385 x^4 \left (2 a c+b^2\right )-2310 a b x^2+330 b c x^6+105 c^2 x^8\right )}{1155 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-231*a^2 - 2310*a*b*x^2 + 385*(b^2 + 2*a*c)*x^4 + 330*b*c*x^6 + 105*c^2*x^8))/(1155*x^(5/2))

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Maple [A]  time = 0.044, size = 49, normalized size = 0.8 \begin{align*} -{\frac{-210\,{c}^{2}{x}^{8}-660\,bc{x}^{6}-1540\,{x}^{4}ac-770\,{b}^{2}{x}^{4}+4620\,ab{x}^{2}+462\,{a}^{2}}{1155}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1155*(-105*c^2*x^8-330*b*c*x^6-770*a*c*x^4-385*b^2*x^4+2310*a*b*x^2+231*a^2)/x^(5/2)

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Maxima [A]  time = 0.969837, size = 61, normalized size = 0.98 \begin{align*} \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b c x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} + 2 \, a c\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*c^2*x^(11/2) + 4/7*b*c*x^(7/2) + 2/3*(b^2 + 2*a*c)*x^(3/2) - 2/5*(10*a*b*x^2 + a^2)/x^(5/2)

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Fricas [A]  time = 1.30901, size = 126, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (105 \, c^{2} x^{8} + 330 \, b c x^{6} + 385 \,{\left (b^{2} + 2 \, a c\right )} x^{4} - 2310 \, a b x^{2} - 231 \, a^{2}\right )}}{1155 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*c^2*x^8 + 330*b*c*x^6 + 385*(b^2 + 2*a*c)*x^4 - 2310*a*b*x^2 - 231*a^2)/x^(5/2)

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Sympy [A]  time = 8.94939, size = 68, normalized size = 1.1 \begin{align*} - \frac{2 a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a b}{\sqrt{x}} + \frac{4 a c x^{\frac{3}{2}}}{3} + \frac{2 b^{2} x^{\frac{3}{2}}}{3} + \frac{4 b c x^{\frac{7}{2}}}{7} + \frac{2 c^{2} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/(5*x**(5/2)) - 4*a*b/sqrt(x) + 4*a*c*x**(3/2)/3 + 2*b**2*x**(3/2)/3 + 4*b*c*x**(7/2)/7 + 2*c**2*x**(11
/2)/11

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Giac [A]  time = 1.1804, size = 63, normalized size = 1.02 \begin{align*} \frac{2}{11} \, c^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b c x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + \frac{4}{3} \, a c x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b x^{2} + a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*c^2*x^(11/2) + 4/7*b*c*x^(7/2) + 2/3*b^2*x^(3/2) + 4/3*a*c*x^(3/2) - 2/5*(10*a*b*x^2 + a^2)/x^(5/2)