3.2.81 \(\int \frac {(A+B x^2) (b x^2+c x^4)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac {2}{9} A b^3 x^{9/2}+\frac {2}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{25} B c^3 x^{25/2} \]

________________________________________________________________________________________

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}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{9} A b^3 x^{9/2}+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{25} B c^3 x^{25/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(5/2),x]

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (6*b*c*(b*B + A*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*
c)*x^(21/2))/21 + (2*B*c^3*x^(25/2))/25

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^{5/2}} \, dx &=\int x^{7/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{7/2}+b^2 (b B+3 A c) x^{11/2}+3 b c (b B+A c) x^{15/2}+c^2 (3 b B+A c) x^{19/2}+B c^3 x^{23/2}\right ) \, dx\\ &=\frac {2}{9} A b^3 x^{9/2}+\frac {2}{13} b^2 (b B+3 A c) x^{13/2}+\frac {6}{17} b c (b B+A c) x^{17/2}+\frac {2}{21} c^2 (3 b B+A c) x^{21/2}+\frac {2}{25} B c^3 x^{25/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{9} A b^3 x^{9/2}+\frac {2}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{25} B c^3 x^{25/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(5/2),x]

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (6*b*c*(b*B + A*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*
c)*x^(21/2))/21 + (2*B*c^3*x^(25/2))/25

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.06, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (38675 A b^3 x^{9/2}+80325 A b^2 c x^{13/2}+61425 A b c^2 x^{17/2}+16575 A c^3 x^{21/2}+26775 b^3 B x^{13/2}+61425 b^2 B c x^{17/2}+49725 b B c^2 x^{21/2}+13923 B c^3 x^{25/2}\right )}{348075} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(5/2),x]

[Out]

(2*(38675*A*b^3*x^(9/2) + 26775*b^3*B*x^(13/2) + 80325*A*b^2*c*x^(13/2) + 61425*b^2*B*c*x^(17/2) + 61425*A*b*c
^2*x^(17/2) + 49725*b*B*c^2*x^(21/2) + 16575*A*c^3*x^(21/2) + 13923*B*c^3*x^(25/2)))/348075

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{348075} \, {\left (13923 \, B c^{3} x^{12} + 16575 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 61425 \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + 38675 \, A b^{3} x^{4} + 26775 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6}\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^(5/2),x, algorithm="fricas")

[Out]

2/348075*(13923*B*c^3*x^12 + 16575*(3*B*b*c^2 + A*c^3)*x^10 + 61425*(B*b^2*c + A*b*c^2)*x^8 + 38675*A*b^3*x^4
+ 26775*(B*b^3 + 3*A*b^2*c)*x^6)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{25} \, B c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, B b c^{2} x^{\frac {21}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, B b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, A b c^{2} x^{\frac {17}{2}} + \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {6}{13} \, A b^{2} c x^{\frac {13}{2}} + \frac {2}{9} \, A b^{3} 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)^3/x^(5/2),x, algorithm="giac")

[Out]

2/25*B*c^3*x^(25/2) + 2/7*B*b*c^2*x^(21/2) + 2/21*A*c^3*x^(21/2) + 6/17*B*b^2*c*x^(17/2) + 6/17*A*b*c^2*x^(17/
2) + 2/13*B*b^3*x^(13/2) + 6/13*A*b^2*c*x^(13/2) + 2/9*A*b^3*x^(9/2)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (13923 B \,c^{3} x^{8}+16575 A \,c^{3} x^{6}+49725 B b \,c^{2} x^{6}+61425 A b \,c^{2} x^{4}+61425 B \,b^{2} c \,x^{4}+80325 A \,b^{2} c \,x^{2}+26775 B \,b^{3} x^{2}+38675 A \,b^{3}\right ) x^{\frac {9}{2}}}{348075} \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^(5/2),x)

[Out]

2/348075*x^(9/2)*(13923*B*c^3*x^8+16575*A*c^3*x^6+49725*B*b*c^2*x^6+61425*A*b*c^2*x^4+61425*B*b^2*c*x^4+80325*
A*b^2*c*x^2+26775*B*b^3*x^2+38675*A*b^3)

________________________________________________________________________________________

maxima [A]  time = 1.40, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{25} \, B c^{3} x^{\frac {25}{2}} + \frac {2}{21} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {21}{2}} + \frac {6}{17} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {13}{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^(5/2),x, algorithm="maxima")

[Out]

2/25*B*c^3*x^(25/2) + 2/21*(3*B*b*c^2 + A*c^3)*x^(21/2) + 6/17*(B*b^2*c + A*b*c^2)*x^(17/2) + 2/9*A*b^3*x^(9/2
) + 2/13*(B*b^3 + 3*A*b^2*c)*x^(13/2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 69, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,B\,b^3}{13}+\frac {6\,A\,c\,b^2}{13}\right )+x^{21/2}\,\left (\frac {2\,A\,c^3}{21}+\frac {2\,B\,b\,c^2}{7}\right )+\frac {2\,A\,b^3\,x^{9/2}}{9}+\frac {2\,B\,c^3\,x^{25/2}}{25}+\frac {6\,b\,c\,x^{17/2}\,\left (A\,c+B\,b\right )}{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)^3)/x^(5/2),x)

[Out]

x^(13/2)*((2*B*b^3)/13 + (6*A*b^2*c)/13) + x^(21/2)*((2*A*c^3)/21 + (2*B*b*c^2)/7) + (2*A*b^3*x^(9/2))/9 + (2*
B*c^3*x^(25/2))/25 + (6*b*c*x^(17/2)*(A*c + B*b))/17

________________________________________________________________________________________

sympy [A]  time = 35.91, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {9}{2}}}{9} + \frac {6 A b^{2} c x^{\frac {13}{2}}}{13} + \frac {6 A b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A c^{3} x^{\frac {21}{2}}}{21} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} + \frac {6 B b^{2} c x^{\frac {17}{2}}}{17} + \frac {2 B b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 B c^{3} x^{\frac {25}{2}}}{25} \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**(5/2),x)

[Out]

2*A*b**3*x**(9/2)/9 + 6*A*b**2*c*x**(13/2)/13 + 6*A*b*c**2*x**(17/2)/17 + 2*A*c**3*x**(21/2)/21 + 2*B*b**3*x**
(13/2)/13 + 6*B*b**2*c*x**(17/2)/17 + 2*B*b*c**2*x**(21/2)/7 + 2*B*c**3*x**(25/2)/25

________________________________________________________________________________________