3.1058 \(\int \frac{(a+b x^2+c x^4)^3}{\sqrt{x}} \, dx\)

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

[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

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Rubi [A]  time = 0.0437411, antiderivative size = 101, 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} \[ \frac{6}{5} a^2 b x^{5/2}+2 a^3 \sqrt{x}+\frac{6}{17} c x^{17/2} \left (a c+b^2\right )+\frac{2}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac{2}{3} a x^{9/2} \left (a c+b^2\right )+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2} \]

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*}

Mathematica [A]  time = 0.0706392, size = 102, normalized size = 1.01 \[ 2 \left (\frac{3}{5} a^2 b x^{5/2}+a^3 \sqrt{x}+\frac{3}{17} c x^{17/2} \left (a c+b^2\right )+\frac{1}{13} b x^{13/2} \left (6 a c+b^2\right )+\frac{1}{3} a x^{9/2} \left (a c+b^2\right )+\frac{1}{7} b c^2 x^{21/2}+\frac{1}{25} c^3 x^{25/2}\right ) \]

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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Maple [A]  time = 0.045, size = 90, normalized size = 0.9 \begin{align*}{\frac{9282\,{c}^{3}{x}^{12}+33150\,b{c}^{2}{x}^{10}+40950\,{x}^{8}a{c}^{2}+40950\,{x}^{8}{b}^{2}c+107100\,{x}^{6}abc+17850\,{x}^{6}{b}^{3}+77350\,{a}^{2}c{x}^{4}+77350\,{x}^{4}{b}^{2}a+139230\,{a}^{2}b{x}^{2}+232050\,{a}^{3}}{116025}\sqrt{x}} \end{align*}

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 = 0.967681, size = 119, normalized size = 1.18 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.23518, size = 225, normalized size = 2.23 \begin{align*} \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{align*}

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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Sympy [A]  time = 19.3961, size = 128, normalized size = 1.27 \begin{align*} 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{align*}

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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Giac [A]  time = 1.13289, size = 117, normalized size = 1.16 \begin{align*} \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{align*}

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)