3.2.82 \(\int \frac {(A+B x^2) (b x^2+c x^4)^3}{x^{7/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac {2}{7} A b^3 x^{7/2}+\frac {2}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{23} B c^3 x^{23/2} \]

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Rubi [A]  time = 0.05, antiderivative size = 85, 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 = {1584, 448} \begin {gather*} \frac {2}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{7} A b^3 x^{7/2}+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{23} B c^3 x^{23/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(7/2),x]

[Out]

(2*A*b^3*x^(7/2))/7 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (2*b*c*(b*B + A*c)*x^(15/2))/5 + (2*c^2*(3*b*B + A*c
)*x^(19/2))/19 + (2*B*c^3*x^(23/2))/23

Rule 448

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 1584

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

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Mathematica [A]  time = 0.05, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{7} A b^3 x^{7/2}+\frac {2}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{23} B c^3 x^{23/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(7/2),x]

[Out]

(2*A*b^3*x^(7/2))/7 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (2*b*c*(b*B + A*c)*x^(15/2))/5 + (2*c^2*(3*b*B + A*c
)*x^(19/2))/19 + (2*B*c^3*x^(23/2))/23

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IntegrateAlgebraic [A]  time = 0.06, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (24035 A b^3 x^{7/2}+45885 A b^2 c x^{11/2}+33649 A b c^2 x^{15/2}+8855 A c^3 x^{19/2}+15295 b^3 B x^{11/2}+33649 b^2 B c x^{15/2}+26565 b B c^2 x^{19/2}+7315 B c^3 x^{23/2}\right )}{168245} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(7/2),x]

[Out]

(2*(24035*A*b^3*x^(7/2) + 15295*b^3*B*x^(11/2) + 45885*A*b^2*c*x^(11/2) + 33649*b^2*B*c*x^(15/2) + 33649*A*b*c
^2*x^(15/2) + 26565*b*B*c^2*x^(19/2) + 8855*A*c^3*x^(19/2) + 7315*B*c^3*x^(23/2)))/168245

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fricas [A]  time = 0.40, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{168245} \, {\left (7315 \, B c^{3} x^{11} + 8855 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 33649 \, {\left (B b^{2} c + A b c^{2}\right )} x^{7} + 24035 \, A b^{3} x^{3} + 15295 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5}\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)^3/x^(7/2),x, algorithm="fricas")

[Out]

2/168245*(7315*B*c^3*x^11 + 8855*(3*B*b*c^2 + A*c^3)*x^9 + 33649*(B*b^2*c + A*b*c^2)*x^7 + 24035*A*b^3*x^3 + 1
5295*(B*b^3 + 3*A*b^2*c)*x^5)*sqrt(x)

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giac [A]  time = 0.15, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {6}{19} \, B b c^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, B b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, A b c^{2} x^{\frac {15}{2}} + \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A b^{2} c x^{\frac {11}{2}} + \frac {2}{7} \, A b^{3} x^{\frac {7}{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)^3/x^(7/2),x, algorithm="giac")

[Out]

2/23*B*c^3*x^(23/2) + 6/19*B*b*c^2*x^(19/2) + 2/19*A*c^3*x^(19/2) + 2/5*B*b^2*c*x^(15/2) + 2/5*A*b*c^2*x^(15/2
) + 2/11*B*b^3*x^(11/2) + 6/11*A*b^2*c*x^(11/2) + 2/7*A*b^3*x^(7/2)

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maple [A]  time = 0.05, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (7315 B \,c^{3} x^{8}+8855 A \,c^{3} x^{6}+26565 B b \,c^{2} x^{6}+33649 A b \,c^{2} x^{4}+33649 B \,b^{2} c \,x^{4}+45885 A \,b^{2} c \,x^{2}+15295 B \,b^{3} x^{2}+24035 A \,b^{3}\right ) x^{\frac {7}{2}}}{168245} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)^3/x^(7/2),x)

[Out]

2/168245*x^(7/2)*(7315*B*c^3*x^8+8855*A*c^3*x^6+26565*B*b*c^2*x^6+33649*A*b*c^2*x^4+33649*B*b^2*c*x^4+45885*A*
b^2*c*x^2+15295*B*b^3*x^2+24035*A*b^3)

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maxima [A]  time = 1.34, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {19}{2}} + \frac {2}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A b^{3} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {11}{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)^3/x^(7/2),x, algorithm="maxima")

[Out]

2/23*B*c^3*x^(23/2) + 2/19*(3*B*b*c^2 + A*c^3)*x^(19/2) + 2/5*(B*b^2*c + A*b*c^2)*x^(15/2) + 2/7*A*b^3*x^(7/2)
+ 2/11*(B*b^3 + 3*A*b^2*c)*x^(11/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{11/2}\,\left (\frac {2\,B\,b^3}{11}+\frac {6\,A\,c\,b^2}{11}\right )+x^{19/2}\,\left (\frac {2\,A\,c^3}{19}+\frac {6\,B\,b\,c^2}{19}\right )+\frac {2\,A\,b^3\,x^{7/2}}{7}+\frac {2\,B\,c^3\,x^{23/2}}{23}+\frac {2\,b\,c\,x^{15/2}\,\left (A\,c+B\,b\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(7/2),x)

[Out]

x^(11/2)*((2*B*b^3)/11 + (6*A*b^2*c)/11) + x^(19/2)*((2*A*c^3)/19 + (6*B*b*c^2)/19) + (2*A*b^3*x^(7/2))/7 + (2
*B*c^3*x^(23/2))/23 + (2*b*c*x^(15/2)*(A*c + B*b))/5

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sympy [A]  time = 53.84, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {7}{2}}}{7} + \frac {6 A b^{2} c x^{\frac {11}{2}}}{11} + \frac {2 A b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A c^{3} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {11}{2}}}{11} + \frac {2 B b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 B b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 B c^{3} x^{\frac {23}{2}}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)**3/x**(7/2),x)

[Out]

2*A*b**3*x**(7/2)/7 + 6*A*b**2*c*x**(11/2)/11 + 2*A*b*c**2*x**(15/2)/5 + 2*A*c**3*x**(19/2)/19 + 2*B*b**3*x**(
11/2)/11 + 2*B*b**2*c*x**(15/2)/5 + 6*B*b*c**2*x**(19/2)/19 + 2*B*c**3*x**(23/2)/23

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