3.11.50 \(\int \sqrt {x} (a+b x^2+c x^4)^2 \, dx\) [1050]

Optimal. Leaf size=64 \[ \frac {2}{3} a^2 x^{3/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{11} \left (b^2+2 a c\right ) x^{11/2}+\frac {4}{15} b c x^{15/2}+\frac {2}{19} c^2 x^{19/2} \]

[Out]

2/3*a^2*x^(3/2)+4/7*a*b*x^(7/2)+2/11*(2*a*c+b^2)*x^(11/2)+4/15*b*c*x^(15/2)+2/19*c^2*x^(19/2)

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Rubi [A]
time = 0.02, antiderivative size = 64, 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 = {1122} \begin {gather*} \frac {2}{3} a^2 x^{3/2}+\frac {2}{11} x^{11/2} \left (2 a c+b^2\right )+\frac {4}{7} a b x^{7/2}+\frac {4}{15} b c x^{15/2}+\frac {2}{19} c^2 x^{19/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x^2 + c*x^4)^2,x]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*b*x^(7/2))/7 + (2*(b^2 + 2*a*c)*x^(11/2))/11 + (4*b*c*x^(15/2))/15 + (2*c^2*x^(19/2))
/19

Rule 1122

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 \sqrt {x} \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 \sqrt {x}+2 a b x^{5/2}+\left (b^2+2 a c\right ) x^{9/2}+2 b c x^{13/2}+c^2 x^{17/2}\right ) \, dx\\ &=\frac {2}{3} a^2 x^{3/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{11} \left (b^2+2 a c\right ) x^{11/2}+\frac {4}{15} b c x^{15/2}+\frac {2}{19} c^2 x^{19/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.89 \begin {gather*} \frac {2 x^{3/2} \left (7315 a^2+570 a \left (11 b x^2+7 c x^4\right )+7 \left (285 b^2 x^4+418 b c x^6+165 c^2 x^8\right )\right )}{21945} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(7315*a^2 + 570*a*(11*b*x^2 + 7*c*x^4) + 7*(285*b^2*x^4 + 418*b*c*x^6 + 165*c^2*x^8)))/21945

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Maple [A]
time = 0.06, size = 45, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {2 \left (2 a c +b^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {4 b c \,x^{\frac {15}{2}}}{15}+\frac {2 c^{2} x^{\frac {19}{2}}}{19}\)	\(45\)
default	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {2 \left (2 a c +b^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {4 b c \,x^{\frac {15}{2}}}{15}+\frac {2 c^{2} x^{\frac {19}{2}}}{19}\)	\(45\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (1155 c^{2} x^{8}+2926 b c \,x^{6}+3990 c \,x^{4} a +1995 b^{2} x^{4}+6270 a b \,x^{2}+7315 a^{2}\right )}{21945}\)	\(49\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (1155 c^{2} x^{8}+2926 b c \,x^{6}+3990 c \,x^{4} a +1995 b^{2} x^{4}+6270 a b \,x^{2}+7315 a^{2}\right )}{21945}\)	\(49\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (1155 c^{2} x^{8}+2926 b c \,x^{6}+3990 c \,x^{4} a +1995 b^{2} x^{4}+6270 a b \,x^{2}+7315 a^{2}\right )}{21945}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/3*a^2*x^(3/2)+4/7*a*b*x^(7/2)+2/11*(2*a*c+b^2)*x^(11/2)+4/15*b*c*x^(15/2)+2/19*c^2*x^(19/2)

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Maxima [A]
time = 0.30, size = 44, normalized size = 0.69 \begin {gather*} \frac {2}{19} \, c^{2} x^{\frac {19}{2}} + \frac {4}{15} \, b c x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{2} + 2 \, a c\right )} x^{\frac {11}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*c^2*x^(19/2) + 4/15*b*c*x^(15/2) + 2/11*(b^2 + 2*a*c)*x^(11/2) + 4/7*a*b*x^(7/2) + 2/3*a^2*x^(3/2)

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Fricas [A]
time = 0.32, size = 47, normalized size = 0.73 \begin {gather*} \frac {2}{21945} \, {\left (1155 \, c^{2} x^{9} + 2926 \, b c x^{7} + 1995 \, {\left (b^{2} + 2 \, a c\right )} x^{5} + 6270 \, a b x^{3} + 7315 \, a^{2} x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21945*(1155*c^2*x^9 + 2926*b*c*x^7 + 1995*(b^2 + 2*a*c)*x^5 + 6270*a*b*x^3 + 7315*a^2*x)*sqrt(x)

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Sympy [A]
time = 1.56, size = 63, normalized size = 0.98 \begin {gather*} \frac {2 a^{2} x^{\frac {3}{2}}}{3} + \frac {4 a b x^{\frac {7}{2}}}{7} + \frac {4 b c x^{\frac {15}{2}}}{15} + \frac {2 c^{2} x^{\frac {19}{2}}}{19} + \frac {2 x^{\frac {11}{2}} \cdot \left (2 a c + b^{2}\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*b*x**(7/2)/7 + 4*b*c*x**(15/2)/15 + 2*c**2*x**(19/2)/19 + 2*x**(11/2)*(2*a*c + b**2)/1
1

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Giac [A]
time = 3.28, size = 46, normalized size = 0.72 \begin {gather*} \frac {2}{19} \, c^{2} x^{\frac {19}{2}} + \frac {4}{15} \, b c x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} x^{\frac {11}{2}} + \frac {4}{11} \, a c x^{\frac {11}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/19*c^2*x^(19/2) + 4/15*b*c*x^(15/2) + 2/11*b^2*x^(11/2) + 4/11*a*c*x^(11/2) + 4/7*a*b*x^(7/2) + 2/3*a^2*x^(3
/2)

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Mupad [B]
time = 0.03, size = 45, normalized size = 0.70 \begin {gather*} x^{11/2}\,\left (\frac {2\,b^2}{11}+\frac {4\,a\,c}{11}\right )+\frac {2\,a^2\,x^{3/2}}{3}+\frac {2\,c^2\,x^{19/2}}{19}+\frac {4\,a\,b\,x^{7/2}}{7}+\frac {4\,b\,c\,x^{15/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x^2 + c*x^4)^2,x)

[Out]

x^(11/2)*((4*a*c)/11 + (2*b^2)/11) + (2*a^2*x^(3/2))/3 + (2*c^2*x^(19/2))/19 + (4*a*b*x^(7/2))/7 + (4*b*c*x^(1
5/2))/15

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