3.11.52 \(\int \frac {(a+b x^2+c x^4)^2}{x^{3/2}} \, dx\) [1052]

Optimal. Leaf size=62 \[ -\frac {2 a^2}{\sqrt {x}}+\frac {4}{3} a b x^{3/2}+\frac {2}{7} \left (b^2+2 a c\right ) x^{7/2}+\frac {4}{11} b c x^{11/2}+\frac {2}{15} c^2 x^{15/2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 62, 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 a^2}{\sqrt {x}}+\frac {2}{7} x^{7/2} \left (2 a c+b^2\right )+\frac {4}{3} a b x^{3/2}+\frac {4}{11} b c x^{11/2}+\frac {2}{15} c^2 x^{15/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*(b^2 + 2*a*c)*x^(7/2))/7 + (4*b*c*x^(11/2))/11 + (2*c^2*x^(15/2))/15

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

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Mathematica [A]
time = 0.03, size = 52, normalized size = 0.84 \begin {gather*} -\frac {2 \left (1155 a^2-770 a b x^2-165 b^2 x^4-330 a c x^4-210 b c x^6-77 c^2 x^8\right )}{1155 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1155*a^2 - 770*a*b*x^2 - 165*b^2*x^4 - 330*a*c*x^4 - 210*b*c*x^6 - 77*c^2*x^8))/(1155*Sqrt[x])

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Maple [A]
time = 0.04, size = 47, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {2 c^{2} x^{\frac {15}{2}}}{15}+\frac {4 b c \,x^{\frac {11}{2}}}{11}+\frac {4 a c \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} x^{\frac {7}{2}}}{7}+\frac {4 a b \,x^{\frac {3}{2}}}{3}-\frac {2 a^{2}}{\sqrt {x}}\)	\(47\)
default	\(\frac {2 c^{2} x^{\frac {15}{2}}}{15}+\frac {4 b c \,x^{\frac {11}{2}}}{11}+\frac {4 a c \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} x^{\frac {7}{2}}}{7}+\frac {4 a b \,x^{\frac {3}{2}}}{3}-\frac {2 a^{2}}{\sqrt {x}}\)	\(47\)
gosper	\(-\frac {2 \left (-77 c^{2} x^{8}-210 b c \,x^{6}-330 c \,x^{4} a -165 b^{2} x^{4}-770 a b \,x^{2}+1155 a^{2}\right )}{1155 \sqrt {x}}\)	\(49\)
trager	\(-\frac {2 \left (-77 c^{2} x^{8}-210 b c \,x^{6}-330 c \,x^{4} a -165 b^{2} x^{4}-770 a b \,x^{2}+1155 a^{2}\right )}{1155 \sqrt {x}}\)	\(49\)
risch	\(-\frac {2 \left (-77 c^{2} x^{8}-210 b c \,x^{6}-330 c \,x^{4} a -165 b^{2} x^{4}-770 a b \,x^{2}+1155 a^{2}\right )}{1155 \sqrt {x}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 44, normalized size = 0.71 \begin {gather*} \frac {2}{15} \, c^{2} x^{\frac {15}{2}} + \frac {4}{11} \, b c x^{\frac {11}{2}} + \frac {2}{7} \, {\left (b^{2} + 2 \, a c\right )} x^{\frac {7}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 46, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (77 \, c^{2} x^{8} + 210 \, b c x^{6} + 165 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 770 \, a b x^{2} - 1155 \, a^{2}\right )}}{1155 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(77*c^2*x^8 + 210*b*c*x^6 + 165*(b^2 + 2*a*c)*x^4 + 770*a*b*x^2 - 1155*a^2)/sqrt(x)

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Sympy [A]
time = 0.63, size = 68, normalized size = 1.10 \begin {gather*} - \frac {2 a^{2}}{\sqrt {x}} + \frac {4 a b x^{\frac {3}{2}}}{3} + \frac {4 a c x^{\frac {7}{2}}}{7} + \frac {2 b^{2} x^{\frac {7}{2}}}{7} + \frac {4 b c x^{\frac {11}{2}}}{11} + \frac {2 c^{2} x^{\frac {15}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**2/sqrt(x) + 4*a*b*x**(3/2)/3 + 4*a*c*x**(7/2)/7 + 2*b**2*x**(7/2)/7 + 4*b*c*x**(11/2)/11 + 2*c**2*x**(15
/2)/15

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Giac [A]
time = 4.51, size = 46, normalized size = 0.74 \begin {gather*} \frac {2}{15} \, c^{2} x^{\frac {15}{2}} + \frac {4}{11} \, b c x^{\frac {11}{2}} + \frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{7} \, a c x^{\frac {7}{2}} + \frac {4}{3} \, a b x^{\frac {3}{2}} - \frac {2 \, a^{2}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.03, size = 45, normalized size = 0.73 \begin {gather*} x^{7/2}\,\left (\frac {2\,b^2}{7}+\frac {4\,a\,c}{7}\right )-\frac {2\,a^2}{\sqrt {x}}+\frac {2\,c^2\,x^{15/2}}{15}+\frac {4\,a\,b\,x^{3/2}}{3}+\frac {4\,b\,c\,x^{11/2}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^2/x^(3/2),x)

[Out]

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

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