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

Optimal. Leaf size=99 \[ -\frac {2 a^3}{\sqrt {x}}+2 a^2 b x^{3/2}+\frac {2}{5} c x^{15/2} \left (a c+b^2\right )+\frac {2}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac {6}{7} a x^{7/2} \left (a c+b^2\right )+\frac {6}{19} b c^2 x^{19/2}+\frac {2}{23} c^3 x^{23/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1108} \begin {gather*} 2 a^2 b x^{3/2}-\frac {2 a^3}{\sqrt {x}}+\frac {2}{5} c x^{15/2} \left (a c+b^2\right )+\frac {2}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac {6}{7} a x^{7/2} \left (a c+b^2\right )+\frac {6}{19} b c^2 x^{19/2}+\frac {2}{23} c^3 x^{23/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^(3/2),x]

[Out]

(-2*a^3)/Sqrt[x] + 2*a^2*b*x^(3/2) + (6*a*(b^2 + a*c)*x^(7/2))/7 + (2*b*(b^2 + 6*a*c)*x^(11/2))/11 + (2*c*(b^2
+ a*c)*x^(15/2))/5 + (6*b*c^2*x^(19/2))/19 + (2*c^3*x^(23/2))/23

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 )^3}{x^{3/2}} \, dx &=\int \left (\frac {a^3}{x^{3/2}}+3 a^2 b \sqrt {x}+3 a \left (b^2+a c\right ) x^{5/2}+b \left (b^2+6 a c\right ) x^{9/2}+3 c \left (b^2+a c\right ) x^{13/2}+3 b c^2 x^{17/2}+c^3 x^{21/2}\right ) \, dx\\ &=-\frac {2 a^3}{\sqrt {x}}+2 a^2 b x^{3/2}+\frac {6}{7} a \left (b^2+a c\right ) x^{7/2}+\frac {2}{11} b \left (b^2+6 a c\right ) x^{11/2}+\frac {2}{5} c \left (b^2+a c\right ) x^{15/2}+\frac {6}{19} b c^2 x^{19/2}+\frac {2}{23} c^3 x^{23/2}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 100, normalized size = 1.01 \begin {gather*} 2 \left (-\frac {a^3}{\sqrt {x}}+a^2 b x^{3/2}+\frac {1}{5} c x^{15/2} \left (a c+b^2\right )+\frac {1}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac {3}{7} a x^{7/2} \left (a c+b^2\right )+\frac {3}{19} b c^2 x^{19/2}+\frac {1}{23} c^3 x^{23/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-(a^3/Sqrt[x]) + a^2*b*x^(3/2) + (3*a*(b^2 + a*c)*x^(7/2))/7 + (b*(b^2 + 6*a*c)*x^(11/2))/11 + (c*(b^2 + a*
c)*x^(15/2))/5 + (3*b*c^2*x^(19/2))/19 + (c^3*x^(23/2))/23)

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IntegrateAlgebraic [A]  time = 0.06, size = 93, normalized size = 0.94 \begin {gather*} \frac {2 \left (-168245 a^3+168245 a^2 b x^2+72105 a^2 c x^4+72105 a b^2 x^4+91770 a b c x^6+33649 a c^2 x^8+15295 b^3 x^6+33649 b^2 c x^8+26565 b c^2 x^{10}+7315 c^3 x^{12}\right )}{168245 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-168245*a^3 + 168245*a^2*b*x^2 + 72105*a*b^2*x^4 + 72105*a^2*c*x^4 + 15295*b^3*x^6 + 91770*a*b*c*x^6 + 336
49*b^2*c*x^8 + 33649*a*c^2*x^8 + 26565*b*c^2*x^10 + 7315*c^3*x^12))/(168245*Sqrt[x])

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fricas [A]  time = 0.93, size = 83, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (7315 \, c^{3} x^{12} + 26565 \, b c^{2} x^{10} + 33649 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 15295 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 168245 \, a^{2} b x^{2} + 72105 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 168245 \, a^{3}\right )}}{168245 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/168245*(7315*c^3*x^12 + 26565*b*c^2*x^10 + 33649*(b^2*c + a*c^2)*x^8 + 15295*(b^3 + 6*a*b*c)*x^6 + 168245*a^
2*b*x^2 + 72105*(a*b^2 + a^2*c)*x^4 - 168245*a^3)/sqrt(x)

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giac [A]  time = 0.16, size = 87, normalized size = 0.88 \begin {gather*} \frac {2}{23} \, c^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b c^{2} x^{\frac {19}{2}} + \frac {2}{5} \, b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, a c^{2} x^{\frac {15}{2}} + \frac {2}{11} \, b^{3} x^{\frac {11}{2}} + \frac {12}{11} \, a b c x^{\frac {11}{2}} + \frac {6}{7} \, a b^{2} x^{\frac {7}{2}} + \frac {6}{7} \, a^{2} c x^{\frac {7}{2}} + 2 \, a^{2} b x^{\frac {3}{2}} - \frac {2 \, a^{3}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*c^3*x^(23/2) + 6/19*b*c^2*x^(19/2) + 2/5*b^2*c*x^(15/2) + 2/5*a*c^2*x^(15/2) + 2/11*b^3*x^(11/2) + 12/11*
a*b*c*x^(11/2) + 6/7*a*b^2*x^(7/2) + 6/7*a^2*c*x^(7/2) + 2*a^2*b*x^(3/2) - 2*a^3/sqrt(x)

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maple [A]  time = 0.01, size = 90, normalized size = 0.91 \begin {gather*} -\frac {2 \left (-7315 c^{3} x^{12}-26565 b \,c^{2} x^{10}-33649 a \,c^{2} x^{8}-33649 b^{2} c \,x^{8}-91770 a b c \,x^{6}-15295 b^{3} x^{6}-72105 a^{2} c \,x^{4}-72105 a \,b^{2} x^{4}-168245 a^{2} b \,x^{2}+168245 a^{3}\right )}{168245 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/168245*(-7315*c^3*x^12-26565*b*c^2*x^10-33649*a*c^2*x^8-33649*b^2*c*x^8-91770*a*b*c*x^6-15295*b^3*x^6-72105
*a^2*c*x^4-72105*a*b^2*x^4-168245*a^2*b*x^2+168245*a^3)/x^(1/2)

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maxima [A]  time = 0.98, size = 81, normalized size = 0.82 \begin {gather*} \frac {2}{23} \, c^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b c^{2} x^{\frac {19}{2}} + \frac {2}{5} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {11}{2}} + 2 \, a^{2} b x^{\frac {3}{2}} + \frac {6}{7} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {7}{2}} - \frac {2 \, a^{3}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*c^3*x^(23/2) + 6/19*b*c^2*x^(19/2) + 2/5*(b^2*c + a*c^2)*x^(15/2) + 2/11*(b^3 + 6*a*b*c)*x^(11/2) + 2*a^2
*b*x^(3/2) + 6/7*(a*b^2 + a^2*c)*x^(7/2) - 2*a^3/sqrt(x)

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mupad [B]  time = 0.04, size = 76, normalized size = 0.77 \begin {gather*} x^{11/2}\,\left (\frac {2\,b^3}{11}+\frac {12\,a\,c\,b}{11}\right )-\frac {2\,a^3}{\sqrt {x}}+\frac {2\,c^3\,x^{23/2}}{23}+2\,a^2\,b\,x^{3/2}+\frac {6\,b\,c^2\,x^{19/2}}{19}+\frac {6\,a\,x^{7/2}\,\left (b^2+a\,c\right )}{7}+\frac {2\,c\,x^{15/2}\,\left (b^2+a\,c\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x^(3/2),x)

[Out]

x^(11/2)*((2*b^3)/11 + (12*a*b*c)/11) - (2*a^3)/x^(1/2) + (2*c^3*x^(23/2))/23 + 2*a^2*b*x^(3/2) + (6*b*c^2*x^(
19/2))/19 + (6*a*x^(7/2)*(a*c + b^2))/7 + (2*c*x^(15/2)*(a*c + b^2))/5

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sympy [A]  time = 19.83, size = 126, normalized size = 1.27 \begin {gather*} - \frac {2 a^{3}}{\sqrt {x}} + 2 a^{2} b x^{\frac {3}{2}} + \frac {6 a^{2} c x^{\frac {7}{2}}}{7} + \frac {6 a b^{2} x^{\frac {7}{2}}}{7} + \frac {12 a b c x^{\frac {11}{2}}}{11} + \frac {2 a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 b^{3} x^{\frac {11}{2}}}{11} + \frac {2 b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 c^{3} x^{\frac {23}{2}}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**3/sqrt(x) + 2*a**2*b*x**(3/2) + 6*a**2*c*x**(7/2)/7 + 6*a*b**2*x**(7/2)/7 + 12*a*b*c*x**(11/2)/11 + 2*a*
c**2*x**(15/2)/5 + 2*b**3*x**(11/2)/11 + 2*b**2*c*x**(15/2)/5 + 6*b*c**2*x**(19/2)/19 + 2*c**3*x**(23/2)/23

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