3.9.27 \(\int \frac {(a+b x^2+c x^4)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac {2 a^3}{3 x^{3/2}}+6 a^2 b \sqrt {x}+\frac {6}{13} c x^{13/2} \left (a c+b^2\right )+\frac {2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac {6}{5} a x^{5/2} \left (a c+b^2\right )+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 101, 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*} 6 a^2 b \sqrt {x}-\frac {2 a^3}{3 x^{3/2}}+\frac {6}{13} c x^{13/2} \left (a c+b^2\right )+\frac {2}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac {6}{5} a x^{5/2} \left (a c+b^2\right )+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) + 6*a^2*b*Sqrt[x] + (6*a*(b^2 + a*c)*x^(5/2))/5 + (2*b*(b^2 + 6*a*c)*x^(9/2))/9 + (6*c*(b
^2 + a*c)*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21/2))/21

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 103, normalized size = 1.02 \begin {gather*} 2 \left (-\frac {a^3}{3 x^{3/2}}+3 a^2 b \sqrt {x}+\frac {3}{13} c x^{13/2} \left (a c+b^2\right )+\frac {1}{9} b x^{9/2} \left (6 a c+b^2\right )+\frac {3}{5} a x^{5/2} \left (a c+b^2\right )+\frac {3}{17} b c^2 x^{17/2}+\frac {1}{21} c^3 x^{21/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

2*(-1/3*a^3/x^(3/2) + 3*a^2*b*Sqrt[x] + (3*a*(b^2 + a*c)*x^(5/2))/5 + (b*(b^2 + 6*a*c)*x^(9/2))/9 + (3*c*(b^2
+ a*c)*x^(13/2))/13 + (3*b*c^2*x^(17/2))/17 + (c^3*x^(21/2))/21)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.06, size = 93, normalized size = 0.92 \begin {gather*} \frac {2 \left (-23205 a^3+208845 a^2 b x^2+41769 a^2 c x^4+41769 a b^2 x^4+46410 a b c x^6+16065 a c^2 x^8+7735 b^3 x^6+16065 b^2 c x^8+12285 b c^2 x^{10}+3315 c^3 x^{12}\right )}{69615 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(2*(-23205*a^3 + 208845*a^2*b*x^2 + 41769*a*b^2*x^4 + 41769*a^2*c*x^4 + 7735*b^3*x^6 + 46410*a*b*c*x^6 + 16065
*b^2*c*x^8 + 16065*a*c^2*x^8 + 12285*b*c^2*x^10 + 3315*c^3*x^12))/(69615*x^(3/2))

________________________________________________________________________________________

fricas [A]  time = 1.14, size = 83, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (3315 \, c^{3} x^{12} + 12285 \, b c^{2} x^{10} + 16065 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 7735 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 208845 \, a^{2} b x^{2} + 41769 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 23205 \, a^{3}\right )}}{69615 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/69615*(3315*c^3*x^12 + 12285*b*c^2*x^10 + 16065*(b^2*c + a*c^2)*x^8 + 7735*(b^3 + 6*a*b*c)*x^6 + 208845*a^2*
b*x^2 + 41769*(a*b^2 + a^2*c)*x^4 - 23205*a^3)/x^(3/2)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 87, normalized size = 0.86 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {6}{13} \, a c^{2} x^{\frac {13}{2}} + \frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {4}{3} \, a b c x^{\frac {9}{2}} + \frac {6}{5} \, a b^{2} x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} c x^{\frac {5}{2}} + 6 \, a^{2} b \sqrt {x} - \frac {2 \, a^{3}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(5/2),x, algorithm="giac")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 6/13*a*c^2*x^(13/2) + 2/9*b^3*x^(9/2) + 4/3*a*
b*c*x^(9/2) + 6/5*a*b^2*x^(5/2) + 6/5*a^2*c*x^(5/2) + 6*a^2*b*sqrt(x) - 2/3*a^3/x^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 90, normalized size = 0.89 \begin {gather*} -\frac {2 \left (-3315 c^{3} x^{12}-12285 b \,c^{2} x^{10}-16065 a \,c^{2} x^{8}-16065 b^{2} c \,x^{8}-46410 a b c \,x^{6}-7735 b^{3} x^{6}-41769 a^{2} c \,x^{4}-41769 a \,b^{2} x^{4}-208845 a^{2} b \,x^{2}+23205 a^{3}\right )}{69615 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^3/x^(5/2),x)

[Out]

-2/69615*(-3315*c^3*x^12-12285*b*c^2*x^10-16065*a*c^2*x^8-16065*b^2*c*x^8-46410*a*b*c*x^6-7735*b^3*x^6-41769*a
^2*c*x^4-41769*a*b^2*x^4-208845*a^2*b*x^2+23205*a^3)/x^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.06, size = 81, normalized size = 0.80 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, {\left (b^{2} c + a c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (b^{3} + 6 \, a b c\right )} x^{\frac {9}{2}} + 6 \, a^{2} b \sqrt {x} + \frac {6}{5} \, {\left (a b^{2} + a^{2} c\right )} x^{\frac {5}{2}} - \frac {2 \, a^{3}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*(b^2*c + a*c^2)*x^(13/2) + 2/9*(b^3 + 6*a*b*c)*x^(9/2) + 6*a^2*
b*sqrt(x) + 6/5*(a*b^2 + a^2*c)*x^(5/2) - 2/3*a^3/x^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 76, normalized size = 0.75 \begin {gather*} x^{9/2}\,\left (\frac {2\,b^3}{9}+\frac {4\,a\,c\,b}{3}\right )-\frac {2\,a^3}{3\,x^{3/2}}+\frac {2\,c^3\,x^{21/2}}{21}+6\,a^2\,b\,\sqrt {x}+\frac {6\,b\,c^2\,x^{17/2}}{17}+\frac {6\,a\,x^{5/2}\,\left (b^2+a\,c\right )}{5}+\frac {6\,c\,x^{13/2}\,\left (b^2+a\,c\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x^(5/2),x)

[Out]

x^(9/2)*((2*b^3)/9 + (4*a*b*c)/3) - (2*a^3)/(3*x^(3/2)) + (2*c^3*x^(21/2))/21 + 6*a^2*b*x^(1/2) + (6*b*c^2*x^(
17/2))/17 + (6*a*x^(5/2)*(a*c + b^2))/5 + (6*c*x^(13/2)*(a*c + b^2))/13

________________________________________________________________________________________

sympy [A]  time = 25.44, size = 128, normalized size = 1.27 \begin {gather*} - \frac {2 a^{3}}{3 x^{\frac {3}{2}}} + 6 a^{2} b \sqrt {x} + \frac {6 a^{2} c x^{\frac {5}{2}}}{5} + \frac {6 a b^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b c x^{\frac {9}{2}}}{3} + \frac {6 a c^{2} x^{\frac {13}{2}}}{13} + \frac {2 b^{3} x^{\frac {9}{2}}}{9} + \frac {6 b^{2} c x^{\frac {13}{2}}}{13} + \frac {6 b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 c^{3} x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**3/x**(5/2),x)

[Out]

-2*a**3/(3*x**(3/2)) + 6*a**2*b*sqrt(x) + 6*a**2*c*x**(5/2)/5 + 6*a*b**2*x**(5/2)/5 + 4*a*b*c*x**(9/2)/3 + 6*a
*c**2*x**(13/2)/13 + 2*b**3*x**(9/2)/9 + 6*b**2*c*x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(21/2)/21

________________________________________________________________________________________