3.1061 \(\int \frac {(a+b x^2+c x^4)^3}{x^{7/2}} \, dx\)

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

[Out]

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

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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} \[ -\frac {6 a^2 b}{\sqrt {x}}-\frac {2 a^3}{5 x^{5/2}}+\frac {6}{11} c x^{11/2} \left (a c+b^2\right )+\frac {2}{7} b x^{7/2} \left (6 a c+b^2\right )+2 a x^{3/2} \left (a c+b^2\right )+\frac {2}{5} b c^2 x^{15/2}+\frac {2}{19} c^3 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^(7/2),x]

[Out]

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

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^3/x^(7/2),x]

[Out]

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

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fricas [A]  time = 0.75, size = 83, normalized size = 0.84 \[ \frac {2 \, {\left (385 \, c^{3} x^{12} + 1463 \, b c^{2} x^{10} + 1995 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 1045 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} - 21945 \, a^{2} b x^{2} + 7315 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 1463 \, a^{3}\right )}}{7315 \, x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="fricas")

[Out]

2/7315*(385*c^3*x^12 + 1463*b*c^2*x^10 + 1995*(b^2*c + a*c^2)*x^8 + 1045*(b^3 + 6*a*b*c)*x^6 - 21945*a^2*b*x^2
+ 7315*(a*b^2 + a^2*c)*x^4 - 1463*a^3)/x^(5/2)

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giac [A]  time = 0.17, size = 88, normalized size = 0.89 \[ \frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b c^{2} x^{\frac {15}{2}} + \frac {6}{11} \, b^{2} c x^{\frac {11}{2}} + \frac {6}{11} \, a c^{2} x^{\frac {11}{2}} + \frac {2}{7} \, b^{3} x^{\frac {7}{2}} + \frac {12}{7} \, a b c x^{\frac {7}{2}} + 2 \, a b^{2} x^{\frac {3}{2}} + 2 \, a^{2} c x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 90, normalized size = 0.91 \[ -\frac {2 \left (-385 c^{3} x^{12}-1463 b \,c^{2} x^{10}-1995 a \,c^{2} x^{8}-1995 b^{2} c \,x^{8}-6270 a b c \,x^{6}-1045 b^{3} x^{6}-7315 a^{2} c \,x^{4}-7315 a \,b^{2} x^{4}+21945 a^{2} b \,x^{2}+1463 a^{3}\right )}{7315 x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^3/x^(7/2),x)

[Out]

-2/7315*(-385*c^3*x^12-1463*b*c^2*x^10-1995*a*c^2*x^8-1995*b^2*c*x^8-6270*a*b*c*x^6-1045*b^3*x^6-7315*a^2*c*x^
4-7315*a*b^2*x^4+21945*a^2*b*x^2+1463*a^3)/x^(5/2)

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maxima [A]  time = 1.05, size = 82, normalized size = 0.83 \[ \frac {2}{19} \, c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b c^{2} x^{\frac {15}{2}} + \frac {6}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {7}{2}} + 2 \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 79, normalized size = 0.80 \[ x^{7/2}\,\left (\frac {2\,b^3}{7}+\frac {12\,a\,c\,b}{7}\right )-\frac {\frac {2\,a^3}{5}+6\,b\,a^2\,x^2}{x^{5/2}}+\frac {2\,c^3\,x^{19/2}}{19}+\frac {2\,b\,c^2\,x^{15/2}}{5}+2\,a\,x^{3/2}\,\left (b^2+a\,c\right )+\frac {6\,c\,x^{11/2}\,\left (b^2+a\,c\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x^(7/2),x)

[Out]

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

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sympy [A]  time = 31.86, size = 124, normalized size = 1.25 \[ - \frac {2 a^{3}}{5 x^{\frac {5}{2}}} - \frac {6 a^{2} b}{\sqrt {x}} + 2 a^{2} c x^{\frac {3}{2}} + 2 a b^{2} x^{\frac {3}{2}} + \frac {12 a b c x^{\frac {7}{2}}}{7} + \frac {6 a c^{2} x^{\frac {11}{2}}}{11} + \frac {2 b^{3} x^{\frac {7}{2}}}{7} + \frac {6 b^{2} c x^{\frac {11}{2}}}{11} + \frac {2 b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 c^{3} x^{\frac {19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**3/x**(7/2),x)

[Out]

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

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