3.2.73 \(\int \frac {(A+B x^2) (b x^2+c x^4)^2}{x^{5/2}} \, dx\) [173]

Optimal. Leaf size=63 \[ \frac {2}{5} A b^2 x^{5/2}+\frac {2}{9} b (b B+2 A c) x^{9/2}+\frac {2}{13} c (2 b B+A c) x^{13/2}+\frac {2}{17} B c^2 x^{17/2} \]

[Out]

2/5*A*b^2*x^(5/2)+2/9*b*(2*A*c+B*b)*x^(9/2)+2/13*c*(A*c+2*B*b)*x^(13/2)+2/17*B*c^2*x^(17/2)

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Rubi [A]
time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1598, 459} \begin {gather*} \frac {2}{5} A b^2 x^{5/2}+\frac {2}{13} c x^{13/2} (A c+2 b B)+\frac {2}{9} b x^{9/2} (2 A c+b B)+\frac {2}{17} B c^2 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(5/2),x]

[Out]

(2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*c)*x^(9/2))/9 + (2*c*(2*b*B + A*c)*x^(13/2))/13 + (2*B*c^2*x^(17/2))/17

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.97 \begin {gather*} \frac {2 x^{5/2} \left (17 A \left (117 b^2+130 b c x^2+45 c^2 x^4\right )+5 B x^2 \left (221 b^2+306 b c x^2+117 c^2 x^4\right )\right )}{9945} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(17*A*(117*b^2 + 130*b*c*x^2 + 45*c^2*x^4) + 5*B*x^2*(221*b^2 + 306*b*c*x^2 + 117*c^2*x^4)))/9945

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Maple [A]
time = 0.36, size = 52, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {2 B \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,c^{2}+2 b B c \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 A b c +b^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 A \,b^{2} x^{\frac {5}{2}}}{5}\)	\(52\)
default	\(\frac {2 B \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,c^{2}+2 b B c \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 A b c +b^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 A \,b^{2} x^{\frac {5}{2}}}{5}\)	\(52\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (585 B \,c^{2} x^{6}+765 A \,c^{2} x^{4}+1530 x^{4} b B c +2210 A b c \,x^{2}+1105 b^{2} B \,x^{2}+1989 b^{2} A \right )}{9945}\)	\(56\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (585 B \,c^{2} x^{6}+765 A \,c^{2} x^{4}+1530 x^{4} b B c +2210 A b c \,x^{2}+1105 b^{2} B \,x^{2}+1989 b^{2} A \right )}{9945}\)	\(56\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (585 B \,c^{2} x^{6}+765 A \,c^{2} x^{4}+1530 x^{4} b B c +2210 A b c \,x^{2}+1105 b^{2} B \,x^{2}+1989 b^{2} A \right )}{9945}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)^2/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/17*B*c^2*x^(17/2)+2/13*(A*c^2+2*B*b*c)*x^(13/2)+2/9*(2*A*b*c+B*b^2)*x^(9/2)+2/5*A*b^2*x^(5/2)

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Maxima [A]
time = 0.26, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + \frac {2}{9} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^2*x^(17/2) + 2/13*(2*B*b*c + A*c^2)*x^(13/2) + 2/5*A*b^2*x^(5/2) + 2/9*(B*b^2 + 2*A*b*c)*x^(9/2)

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Fricas [A]
time = 1.55, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{9945} \, {\left (585 \, B c^{2} x^{8} + 765 \, {\left (2 \, B b c + A c^{2}\right )} x^{6} + 1989 \, A b^{2} x^{2} + 1105 \, {\left (B b^{2} + 2 \, A b c\right )} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9945*(585*B*c^2*x^8 + 765*(2*B*b*c + A*c^2)*x^6 + 1989*A*b^2*x^2 + 1105*(B*b^2 + 2*A*b*c)*x^4)*sqrt(x)

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Sympy [A]
time = 1.02, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 A b^{2} x^{\frac {5}{2}}}{5} + \frac {4 A b c x^{\frac {9}{2}}}{9} + \frac {2 A c^{2} x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} x^{\frac {9}{2}}}{9} + \frac {4 B b c x^{\frac {13}{2}}}{13} + \frac {2 B c^{2} x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)**2/x**(5/2),x)

[Out]

2*A*b**2*x**(5/2)/5 + 4*A*b*c*x**(9/2)/9 + 2*A*c**2*x**(13/2)/13 + 2*B*b**2*x**(9/2)/9 + 4*B*b*c*x**(13/2)/13
+ 2*B*c**2*x**(17/2)/17

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Giac [A]
time = 0.43, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{17} \, B c^{2} x^{\frac {17}{2}} + \frac {4}{13} \, B b c x^{\frac {13}{2}} + \frac {2}{13} \, A c^{2} x^{\frac {13}{2}} + \frac {2}{9} \, B b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, A b c x^{\frac {9}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^2*x^(17/2) + 4/13*B*b*c*x^(13/2) + 2/13*A*c^2*x^(13/2) + 2/9*B*b^2*x^(9/2) + 4/9*A*b*c*x^(9/2) + 2/5*
A*b^2*x^(5/2)

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Mupad [B]
time = 0.05, size = 51, normalized size = 0.81 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,b^2}{9}+\frac {4\,A\,c\,b}{9}\right )+x^{13/2}\,\left (\frac {2\,A\,c^2}{13}+\frac {4\,B\,b\,c}{13}\right )+\frac {2\,A\,b^2\,x^{5/2}}{5}+\frac {2\,B\,c^2\,x^{17/2}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(5/2),x)

[Out]

x^(9/2)*((2*B*b^2)/9 + (4*A*b*c)/9) + x^(13/2)*((2*A*c^2)/13 + (4*B*b*c)/13) + (2*A*b^2*x^(5/2))/5 + (2*B*c^2*
x^(17/2))/17

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