3.2.78 \(\int \sqrt {x} (A+B x^2) (b x^2+c x^4)^3 \, dx\)

Optimal. Leaf size=85 \[ \frac {2}{15} A b^3 x^{15/2}+\frac {2}{19} b^2 x^{19/2} (3 A c+b B)+\frac {2}{27} c^2 x^{27/2} (A c+3 b B)+\frac {6}{23} b c x^{23/2} (A c+b B)+\frac {2}{31} B c^3 x^{31/2} \]

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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}{19} b^2 x^{19/2} (3 A c+b B)+\frac {2}{15} A b^3 x^{15/2}+\frac {2}{27} c^2 x^{27/2} (A c+3 b B)+\frac {6}{23} b c x^{23/2} (A c+b B)+\frac {2}{31} B c^3 x^{31/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(19/2))/19 + (6*b*c*(b*B + A*c)*x^(23/2))/23 + (2*c^2*(3*b*B +
A*c)*x^(27/2))/27 + (2*B*c^3*x^(31/2))/31

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 \sqrt {x} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^3 \, dx &=\int x^{13/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{13/2}+b^2 (b B+3 A c) x^{17/2}+3 b c (b B+A c) x^{21/2}+c^2 (3 b B+A c) x^{25/2}+B c^3 x^{29/2}\right ) \, dx\\ &=\frac {2}{15} A b^3 x^{15/2}+\frac {2}{19} b^2 (b B+3 A c) x^{19/2}+\frac {6}{23} b c (b B+A c) x^{23/2}+\frac {2}{27} c^2 (3 b B+A c) x^{27/2}+\frac {2}{31} B c^3 x^{31/2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{15} A b^3 x^{15/2}+\frac {2}{19} b^2 x^{19/2} (3 A c+b B)+\frac {2}{27} c^2 x^{27/2} (A c+3 b B)+\frac {6}{23} b c x^{23/2} (A c+b B)+\frac {2}{31} B c^3 x^{31/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(19/2))/19 + (6*b*c*(b*B + A*c)*x^(23/2))/23 + (2*c^2*(3*b*B +
A*c)*x^(27/2))/27 + (2*B*c^3*x^(31/2))/31

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IntegrateAlgebraic [A]  time = 0.06, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (121923 A b^3 x^{15/2}+288765 A b^2 c x^{19/2}+238545 A b c^2 x^{23/2}+67735 A c^3 x^{27/2}+96255 b^3 B x^{19/2}+238545 b^2 B c x^{23/2}+203205 b B c^2 x^{27/2}+58995 B c^3 x^{31/2}\right )}{1828845} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(121923*A*b^3*x^(15/2) + 96255*b^3*B*x^(19/2) + 288765*A*b^2*c*x^(19/2) + 238545*b^2*B*c*x^(23/2) + 238545*
A*b*c^2*x^(23/2) + 203205*b*B*c^2*x^(27/2) + 67735*A*c^3*x^(27/2) + 58995*B*c^3*x^(31/2)))/1828845

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fricas [A]  time = 0.39, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{1828845} \, {\left (58995 \, B c^{3} x^{15} + 67735 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + 238545 \, {\left (B b^{2} c + A b c^{2}\right )} x^{11} + 121923 \, A b^{3} x^{7} + 96255 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{9}\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^(1/2),x, algorithm="fricas")

[Out]

2/1828845*(58995*B*c^3*x^15 + 67735*(3*B*b*c^2 + A*c^3)*x^13 + 238545*(B*b^2*c + A*b*c^2)*x^11 + 121923*A*b^3*
x^7 + 96255*(B*b^3 + 3*A*b^2*c)*x^9)*sqrt(x)

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giac [A]  time = 0.17, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{31} \, B c^{3} x^{\frac {31}{2}} + \frac {2}{9} \, B b c^{2} x^{\frac {27}{2}} + \frac {2}{27} \, A c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, B b^{2} c x^{\frac {23}{2}} + \frac {6}{23} \, A b c^{2} x^{\frac {23}{2}} + \frac {2}{19} \, B b^{3} x^{\frac {19}{2}} + \frac {6}{19} \, A b^{2} c x^{\frac {19}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{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^(1/2),x, algorithm="giac")

[Out]

2/31*B*c^3*x^(31/2) + 2/9*B*b*c^2*x^(27/2) + 2/27*A*c^3*x^(27/2) + 6/23*B*b^2*c*x^(23/2) + 6/23*A*b*c^2*x^(23/
2) + 2/19*B*b^3*x^(19/2) + 6/19*A*b^2*c*x^(19/2) + 2/15*A*b^3*x^(15/2)

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maple [A]  time = 0.05, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (58995 B \,c^{3} x^{8}+67735 A \,c^{3} x^{6}+203205 B b \,c^{2} x^{6}+238545 A b \,c^{2} x^{4}+238545 B \,b^{2} c \,x^{4}+288765 A \,b^{2} c \,x^{2}+96255 B \,b^{3} x^{2}+121923 A \,b^{3}\right ) x^{\frac {15}{2}}}{1828845} \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^(1/2),x)

[Out]

2/1828845*x^(15/2)*(58995*B*c^3*x^8+67735*A*c^3*x^6+203205*B*b*c^2*x^6+238545*A*b*c^2*x^4+238545*B*b^2*c*x^4+2
88765*A*b^2*c*x^2+96255*B*b^3*x^2+121923*A*b^3)

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maxima [A]  time = 1.33, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{31} \, B c^{3} x^{\frac {31}{2}} + \frac {2}{27} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {27}{2}} + \frac {6}{23} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {23}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} + \frac {2}{19} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {19}{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^(1/2),x, algorithm="maxima")

[Out]

2/31*B*c^3*x^(31/2) + 2/27*(3*B*b*c^2 + A*c^3)*x^(27/2) + 6/23*(B*b^2*c + A*b*c^2)*x^(23/2) + 2/15*A*b^3*x^(15
/2) + 2/19*(B*b^3 + 3*A*b^2*c)*x^(19/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{19/2}\,\left (\frac {2\,B\,b^3}{19}+\frac {6\,A\,c\,b^2}{19}\right )+x^{27/2}\,\left (\frac {2\,A\,c^3}{27}+\frac {2\,B\,b\,c^2}{9}\right )+\frac {2\,A\,b^3\,x^{15/2}}{15}+\frac {2\,B\,c^3\,x^{31/2}}{31}+\frac {6\,b\,c\,x^{23/2}\,\left (A\,c+B\,b\right )}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(A + B*x^2)*(b*x^2 + c*x^4)^3,x)

[Out]

x^(19/2)*((2*B*b^3)/19 + (6*A*b^2*c)/19) + x^(27/2)*((2*A*c^3)/27 + (2*B*b*c^2)/9) + (2*A*b^3*x^(15/2))/15 + (
2*B*c^3*x^(31/2))/31 + (6*b*c*x^(23/2)*(A*c + B*b))/23

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sympy [A]  time = 5.51, size = 95, normalized size = 1.12 \begin {gather*} \frac {2 A b^{3} x^{\frac {15}{2}}}{15} + \frac {2 B c^{3} x^{\frac {31}{2}}}{31} + \frac {2 x^{\frac {27}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{27} + \frac {2 x^{\frac {23}{2}} \left (3 A b c^{2} + 3 B b^{2} c\right )}{23} + \frac {2 x^{\frac {19}{2}} \left (3 A b^{2} c + B b^{3}\right )}{19} \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**(1/2),x)

[Out]

2*A*b**3*x**(15/2)/15 + 2*B*c**3*x**(31/2)/31 + 2*x**(27/2)*(A*c**3 + 3*B*b*c**2)/27 + 2*x**(23/2)*(3*A*b*c**2
+ 3*B*b**2*c)/23 + 2*x**(19/2)*(3*A*b**2*c + B*b**3)/19

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