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

Optimal. Leaf size=101 \[ 2 a^3 \sqrt {x}+\frac {6}{5} a^2 b x^{5/2}+\frac {6}{17} c x^{17/2} \left (a c+b^2\right )+\frac {2}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac {2}{3} a x^{9/2} \left (a c+b^2\right )+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \]

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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*} \frac {6}{5} a^2 b x^{5/2}+2 a^3 \sqrt {x}+\frac {6}{17} c x^{17/2} \left (a c+b^2\right )+\frac {2}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac {2}{3} a x^{9/2} \left (a c+b^2\right )+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/Sqrt[x],x]

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*(b^2 + a*c)*x^(9/2))/3 + (2*b*(b^2 + 6*a*c)*x^(13/2))/13 + (6*c*(b^
2 + a*c)*x^(17/2))/17 + (2*b*c^2*x^(21/2))/7 + (2*c^3*x^(25/2))/25

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}{\sqrt {x}} \, dx &=\int \left (\frac {a^3}{\sqrt {x}}+3 a^2 b x^{3/2}+3 a \left (b^2+a c\right ) x^{7/2}+b \left (b^2+6 a c\right ) x^{11/2}+3 c \left (b^2+a c\right ) x^{15/2}+3 b c^2 x^{19/2}+c^3 x^{23/2}\right ) \, dx\\ &=2 a^3 \sqrt {x}+\frac {6}{5} a^2 b x^{5/2}+\frac {2}{3} a \left (b^2+a c\right ) x^{9/2}+\frac {2}{13} b \left (b^2+6 a c\right ) x^{13/2}+\frac {6}{17} c \left (b^2+a c\right ) x^{17/2}+\frac {2}{7} b c^2 x^{21/2}+\frac {2}{25} c^3 x^{25/2}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 102, normalized size = 1.01 \begin {gather*} 2 \left (a^3 \sqrt {x}+\frac {3}{5} a^2 b x^{5/2}+\frac {3}{17} c x^{17/2} \left (a c+b^2\right )+\frac {1}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac {1}{3} a x^{9/2} \left (a c+b^2\right )+\frac {1}{7} b c^2 x^{21/2}+\frac {1}{25} c^3 x^{25/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*(a^3*Sqrt[x] + (3*a^2*b*x^(5/2))/5 + (a*(b^2 + a*c)*x^(9/2))/3 + (b*(b^2 + 6*a*c)*x^(13/2))/13 + (3*c*(b^2 +
a*c)*x^(17/2))/17 + (b*c^2*x^(21/2))/7 + (c^3*x^(25/2))/25)

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IntegrateAlgebraic [A]  time = 0.05, size = 111, normalized size = 1.10 \begin {gather*} \frac {2 \left (116025 a^3 \sqrt {x}+69615 a^2 b x^{5/2}+38675 a^2 c x^{9/2}+38675 a b^2 x^{9/2}+53550 a b c x^{13/2}+20475 a c^2 x^{17/2}+8925 b^3 x^{13/2}+20475 b^2 c x^{17/2}+16575 b c^2 x^{21/2}+4641 c^3 x^{25/2}\right )}{116025} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(116025*a^3*Sqrt[x] + 69615*a^2*b*x^(5/2) + 38675*a*b^2*x^(9/2) + 38675*a^2*c*x^(9/2) + 8925*b^3*x^(13/2) +
53550*a*b*c*x^(13/2) + 20475*b^2*c*x^(17/2) + 20475*a*c^2*x^(17/2) + 16575*b*c^2*x^(21/2) + 4641*c^3*x^(25/2)
))/116025

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fricas [A]  time = 1.61, size = 83, normalized size = 0.82 \begin {gather*} \frac {2}{116025} \, {\left (4641 \, c^{3} x^{12} + 16575 \, b c^{2} x^{10} + 20475 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 8925 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} + 69615 \, a^{2} b x^{2} + 38675 \, {\left (a b^{2} + a^{2} c\right )} x^{4} + 116025 \, a^{3}\right )} \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^(1/2),x, algorithm="fricas")

[Out]

2/116025*(4641*c^3*x^12 + 16575*b*c^2*x^10 + 20475*(b^2*c + a*c^2)*x^8 + 8925*(b^3 + 6*a*b*c)*x^6 + 69615*a^2*
b*x^2 + 38675*(a*b^2 + a^2*c)*x^4 + 116025*a^3)*sqrt(x)

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giac [A]  time = 0.17, size = 87, normalized size = 0.86 \begin {gather*} \frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b c^{2} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, a c^{2} x^{\frac {17}{2}} + \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {12}{13} \, a b c x^{\frac {13}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {2}{3} \, a^{2} c x^{\frac {9}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 6/17*a*c^2*x^(17/2) + 2/13*b^3*x^(13/2) + 12/13
*a*b*c*x^(13/2) + 2/3*a*b^2*x^(9/2) + 2/3*a^2*c*x^(9/2) + 6/5*a^2*b*x^(5/2) + 2*a^3*sqrt(x)

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maple [A]  time = 0.01, size = 90, normalized size = 0.89 \begin {gather*} \frac {2 \left (4641 c^{3} x^{12}+16575 b \,c^{2} x^{10}+20475 a \,c^{2} x^{8}+20475 b^{2} c \,x^{8}+53550 a b c \,x^{6}+8925 b^{3} x^{6}+38675 a^{2} c \,x^{4}+38675 a \,b^{2} x^{4}+69615 a^{2} b \,x^{2}+116025 a^{3}\right ) \sqrt {x}}{116025} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^3/x^(1/2),x)

[Out]

2/116025*x^(1/2)*(4641*c^3*x^12+16575*b*c^2*x^10+20475*a*c^2*x^8+20475*b^2*c*x^8+53550*a*b*c*x^6+8925*b^3*x^6+
38675*a^2*c*x^4+38675*a*b^2*x^4+69615*a^2*b*x^2+116025*a^3)

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maxima [A]  time = 1.04, size = 88, normalized size = 0.87 \begin {gather*} \frac {2}{25} \, c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, b c^{2} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + 2 \, a^{3} \sqrt {x} + \frac {2}{15} \, {\left (5 \, c x^{\frac {9}{2}} + 9 \, b x^{\frac {5}{2}}\right )} a^{2} + \frac {2}{663} \, {\left (117 \, c^{2} x^{\frac {17}{2}} + 306 \, b c x^{\frac {13}{2}} + 221 \, b^{2} x^{\frac {9}{2}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 2/13*b^3*x^(13/2) + 2*a^3*sqrt(x) + 2/15*(5*c*x
^(9/2) + 9*b*x^(5/2))*a^2 + 2/663*(117*c^2*x^(17/2) + 306*b*c*x^(13/2) + 221*b^2*x^(9/2))*a

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mupad [B]  time = 0.03, size = 76, normalized size = 0.75 \begin {gather*} x^{13/2}\,\left (\frac {2\,b^3}{13}+\frac {12\,a\,c\,b}{13}\right )+2\,a^3\,\sqrt {x}+\frac {2\,c^3\,x^{25/2}}{25}+\frac {6\,a^2\,b\,x^{5/2}}{5}+\frac {2\,b\,c^2\,x^{21/2}}{7}+\frac {2\,a\,x^{9/2}\,\left (b^2+a\,c\right )}{3}+\frac {6\,c\,x^{17/2}\,\left (b^2+a\,c\right )}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x^(1/2),x)

[Out]

x^(13/2)*((2*b^3)/13 + (12*a*b*c)/13) + 2*a^3*x^(1/2) + (2*c^3*x^(25/2))/25 + (6*a^2*b*x^(5/2))/5 + (2*b*c^2*x
^(21/2))/7 + (2*a*x^(9/2)*(a*c + b^2))/3 + (6*c*x^(17/2)*(a*c + b^2))/17

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sympy [A]  time = 23.50, size = 128, normalized size = 1.27 \begin {gather*} 2 a^{3} \sqrt {x} + \frac {6 a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 a^{2} c x^{\frac {9}{2}}}{3} + \frac {2 a b^{2} x^{\frac {9}{2}}}{3} + \frac {12 a b c x^{\frac {13}{2}}}{13} + \frac {6 a c^{2} x^{\frac {17}{2}}}{17} + \frac {2 b^{3} x^{\frac {13}{2}}}{13} + \frac {6 b^{2} c x^{\frac {17}{2}}}{17} + \frac {2 b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 c^{3} x^{\frac {25}{2}}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**3/x**(1/2),x)

[Out]

2*a**3*sqrt(x) + 6*a**2*b*x**(5/2)/5 + 2*a**2*c*x**(9/2)/3 + 2*a*b**2*x**(9/2)/3 + 12*a*b*c*x**(13/2)/13 + 6*a
*c**2*x**(17/2)/17 + 2*b**3*x**(13/2)/13 + 6*b**2*c*x**(17/2)/17 + 2*b*c**2*x**(21/2)/7 + 2*c**3*x**(25/2)/25

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