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

Optimal. Leaf size=168 \[ \frac {16 b^3 \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{15015 c^5 x^5}-\frac {8 b^2 \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{3003 c^4 x^3}+\frac {2 b \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{429 c^3 x}-\frac {x \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{143 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c} \]

[Out]

16/15015*b^3*(-13*A*c+8*B*b)*(c*x^4+b*x^2)^(5/2)/c^5/x^5-8/3003*b^2*(-13*A*c+8*B*b)*(c*x^4+b*x^2)^(5/2)/c^4/x^
3+2/429*b*(-13*A*c+8*B*b)*(c*x^4+b*x^2)^(5/2)/c^3/x-1/143*(-13*A*c+8*B*b)*x*(c*x^4+b*x^2)^(5/2)/c^2+1/13*B*x^3
*(c*x^4+b*x^2)^(5/2)/c

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Rubi [A]  time = 0.30, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2039, 2016, 2002, 2014} \[ \frac {16 b^3 \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{15015 c^5 x^5}-\frac {8 b^2 \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{3003 c^4 x^3}+\frac {2 b \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{429 c^3 x}-\frac {x \left (b x^2+c x^4\right )^{5/2} (8 b B-13 A c)}{143 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b^3*(8*b*B - 13*A*c)*(b*x^2 + c*x^4)^(5/2))/(15015*c^5*x^5) - (8*b^2*(8*b*B - 13*A*c)*(b*x^2 + c*x^4)^(5/2
))/(3003*c^4*x^3) + (2*b*(8*b*B - 13*A*c)*(b*x^2 + c*x^4)^(5/2))/(429*c^3*x) - ((8*b*B - 13*A*c)*x*(b*x^2 + c*
x^4)^(5/2))/(143*c^2) + (B*x^3*(b*x^2 + c*x^4)^(5/2))/(13*c)

Rule 2002

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j -
1)), x] - Dist[(b*(n*p + n - j + 1))/(a*(j*p + 1)), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j,
n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2039

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(d*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(b*(m + n + p*(j + n) + 1)), x] - Dist[(a*d*(m
+ j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)), Int[(e*x)^m*(a*x^j + b*x^(j + n))^p, x
], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[
m + n + p*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])

Rubi steps

\begin {align*} \int x^4 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {B x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}-\frac {(8 b B-13 A c) \int x^4 \left (b x^2+c x^4\right )^{3/2} \, dx}{13 c}\\ &=-\frac {(8 b B-13 A c) x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}+\frac {(6 b (8 b B-13 A c)) \int x^2 \left (b x^2+c x^4\right )^{3/2} \, dx}{143 c^2}\\ &=\frac {2 b (8 b B-13 A c) \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {(8 b B-13 A c) x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}-\frac {\left (8 b^2 (8 b B-13 A c)\right ) \int \left (b x^2+c x^4\right )^{3/2} \, dx}{429 c^3}\\ &=-\frac {8 b^2 (8 b B-13 A c) \left (b x^2+c x^4\right )^{5/2}}{3003 c^4 x^3}+\frac {2 b (8 b B-13 A c) \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {(8 b B-13 A c) x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}+\frac {\left (16 b^3 (8 b B-13 A c)\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{3003 c^4}\\ &=\frac {16 b^3 (8 b B-13 A c) \left (b x^2+c x^4\right )^{5/2}}{15015 c^5 x^5}-\frac {8 b^2 (8 b B-13 A c) \left (b x^2+c x^4\right )^{5/2}}{3003 c^4 x^3}+\frac {2 b (8 b B-13 A c) \left (b x^2+c x^4\right )^{5/2}}{429 c^3 x}-\frac {(8 b B-13 A c) x \left (b x^2+c x^4\right )^{5/2}}{143 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{5/2}}{13 c}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 113, normalized size = 0.67 \[ \frac {x \left (b+c x^2\right )^3 \left (-16 b^3 c \left (13 A+20 B x^2\right )+40 b^2 c^2 x^2 \left (13 A+14 B x^2\right )-70 b c^3 x^4 \left (13 A+12 B x^2\right )+105 c^4 x^6 \left (13 A+11 B x^2\right )+128 b^4 B\right )}{15015 c^5 \sqrt {x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b + c*x^2)^3*(128*b^4*B + 105*c^4*x^6*(13*A + 11*B*x^2) - 70*b*c^3*x^4*(13*A + 12*B*x^2) + 40*b^2*c^2*x^2*
(13*A + 14*B*x^2) - 16*b^3*c*(13*A + 20*B*x^2)))/(15015*c^5*Sqrt[x^2*(b + c*x^2)])

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fricas [A]  time = 0.87, size = 154, normalized size = 0.92 \[ \frac {{\left (1155 \, B c^{6} x^{12} + 105 \, {\left (14 \, B b c^{5} + 13 \, A c^{6}\right )} x^{10} + 35 \, {\left (B b^{2} c^{4} + 52 \, A b c^{5}\right )} x^{8} + 128 \, B b^{6} - 208 \, A b^{5} c - 5 \, {\left (8 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} x^{6} + 6 \, {\left (8 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{4} - 8 \, {\left (8 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15015 \, c^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15015*(1155*B*c^6*x^12 + 105*(14*B*b*c^5 + 13*A*c^6)*x^10 + 35*(B*b^2*c^4 + 52*A*b*c^5)*x^8 + 128*B*b^6 - 20
8*A*b^5*c - 5*(8*B*b^3*c^3 - 13*A*b^2*c^4)*x^6 + 6*(8*B*b^4*c^2 - 13*A*b^3*c^3)*x^4 - 8*(8*B*b^5*c - 13*A*b^4*
c^2)*x^2)*sqrt(c*x^4 + b*x^2)/(c^5*x)

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giac [A]  time = 0.34, size = 175, normalized size = 1.04 \[ -\frac {16 \, {\left (8 \, B b^{\frac {13}{2}} - 13 \, A b^{\frac {11}{2}} c\right )} \mathrm {sgn}\relax (x)}{15015 \, c^{5}} + \frac {1155 \, {\left (c x^{2} + b\right )}^{\frac {13}{2}} B \mathrm {sgn}\relax (x) - 5460 \, {\left (c x^{2} + b\right )}^{\frac {11}{2}} B b \mathrm {sgn}\relax (x) + 10010 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} B b^{2} \mathrm {sgn}\relax (x) - 8580 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B b^{3} \mathrm {sgn}\relax (x) + 3003 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b^{4} \mathrm {sgn}\relax (x) + 1365 \, {\left (c x^{2} + b\right )}^{\frac {11}{2}} A c \mathrm {sgn}\relax (x) - 5005 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} A b c \mathrm {sgn}\relax (x) + 6435 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} A b^{2} c \mathrm {sgn}\relax (x) - 3003 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A b^{3} c \mathrm {sgn}\relax (x)}{15015 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/15015*(8*B*b^(13/2) - 13*A*b^(11/2)*c)*sgn(x)/c^5 + 1/15015*(1155*(c*x^2 + b)^(13/2)*B*sgn(x) - 5460*(c*x^
2 + b)^(11/2)*B*b*sgn(x) + 10010*(c*x^2 + b)^(9/2)*B*b^2*sgn(x) - 8580*(c*x^2 + b)^(7/2)*B*b^3*sgn(x) + 3003*(
c*x^2 + b)^(5/2)*B*b^4*sgn(x) + 1365*(c*x^2 + b)^(11/2)*A*c*sgn(x) - 5005*(c*x^2 + b)^(9/2)*A*b*c*sgn(x) + 643
5*(c*x^2 + b)^(7/2)*A*b^2*c*sgn(x) - 3003*(c*x^2 + b)^(5/2)*A*b^3*c*sgn(x))/c^5

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maple [A]  time = 0.05, size = 115, normalized size = 0.68 \[ -\frac {\left (c \,x^{2}+b \right ) \left (-1155 B \,x^{8} c^{4}-1365 A \,c^{4} x^{6}+840 B b \,c^{3} x^{6}+910 A b \,c^{3} x^{4}-560 B \,b^{2} c^{2} x^{4}-520 A \,b^{2} c^{2} x^{2}+320 B \,b^{3} c \,x^{2}+208 A \,b^{3} c -128 B \,b^{4}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{15015 c^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15015*(c*x^2+b)*(-1155*B*c^4*x^8-1365*A*c^4*x^6+840*B*b*c^3*x^6+910*A*b*c^3*x^4-560*B*b^2*c^2*x^4-520*A*b^2
*c^2*x^2+320*B*b^3*c*x^2+208*A*b^3*c-128*B*b^4)*(c*x^4+b*x^2)^(3/2)/c^5/x^3

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maxima [A]  time = 1.55, size = 150, normalized size = 0.89 \[ \frac {{\left (105 \, c^{5} x^{10} + 140 \, b c^{4} x^{8} + 5 \, b^{2} c^{3} x^{6} - 6 \, b^{3} c^{2} x^{4} + 8 \, b^{4} c x^{2} - 16 \, b^{5}\right )} \sqrt {c x^{2} + b} A}{1155 \, c^{4}} + \frac {{\left (1155 \, c^{6} x^{12} + 1470 \, b c^{5} x^{10} + 35 \, b^{2} c^{4} x^{8} - 40 \, b^{3} c^{3} x^{6} + 48 \, b^{4} c^{2} x^{4} - 64 \, b^{5} c x^{2} + 128 \, b^{6}\right )} \sqrt {c x^{2} + b} B}{15015 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(105*c^5*x^10 + 140*b*c^4*x^8 + 5*b^2*c^3*x^6 - 6*b^3*c^2*x^4 + 8*b^4*c*x^2 - 16*b^5)*sqrt(c*x^2 + b)*A
/c^4 + 1/15015*(1155*c^6*x^12 + 1470*b*c^5*x^10 + 35*b^2*c^4*x^8 - 40*b^3*c^3*x^6 + 48*b^4*c^2*x^4 - 64*b^5*c*
x^2 + 128*b^6)*sqrt(c*x^2 + b)*B/c^5

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mupad [B]  time = 0.35, size = 143, normalized size = 0.85 \[ \frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {128\,B\,b^6-208\,A\,b^5\,c}{15015\,c^5}+\frac {x^{10}\,\left (1365\,A\,c^6+1470\,B\,b\,c^5\right )}{15015\,c^5}+\frac {B\,c\,x^{12}}{13}+\frac {b^2\,x^6\,\left (13\,A\,c-8\,B\,b\right )}{3003\,c^2}-\frac {2\,b^3\,x^4\,\left (13\,A\,c-8\,B\,b\right )}{5005\,c^3}+\frac {8\,b^4\,x^2\,\left (13\,A\,c-8\,B\,b\right )}{15015\,c^4}+\frac {b\,x^8\,\left (52\,A\,c+B\,b\right )}{429\,c}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x^2 + c*x^4)^(1/2)*((128*B*b^6 - 208*A*b^5*c)/(15015*c^5) + (x^10*(1365*A*c^6 + 1470*B*b*c^5))/(15015*c^5)
+ (B*c*x^12)/13 + (b^2*x^6*(13*A*c - 8*B*b))/(3003*c^2) - (2*b^3*x^4*(13*A*c - 8*B*b))/(5005*c^3) + (8*b^4*x^
2*(13*A*c - 8*B*b))/(15015*c^4) + (b*x^8*(52*A*c + B*b))/(429*c)))/x

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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