3.6.80 \(\int x^5 (a+b x^2)^2 \sqrt {c+d x^2} \, dx\)

Optimal. Leaf size=157 \[ \frac {\left (c+d x^2\right )^{7/2} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{7 d^5}+\frac {c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^5}-\frac {2 b \left (c+d x^2\right )^{9/2} (2 b c-a d)}{9 d^5}-\frac {2 c \left (c+d x^2\right )^{5/2} (b c-a d) (2 b c-a d)}{5 d^5}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^5} \]

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Rubi [A]  time = 0.13, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {446, 88} \begin {gather*} \frac {\left (c+d x^2\right )^{7/2} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{7 d^5}+\frac {c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}{3 d^5}-\frac {2 b \left (c+d x^2\right )^{9/2} (2 b c-a d)}{9 d^5}-\frac {2 c \left (c+d x^2\right )^{5/2} (b c-a d) (2 b c-a d)}{5 d^5}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2)^2*Sqrt[c + d*x^2],x]

[Out]

(c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2))/(3*d^5) - (2*c*(b*c - a*d)*(2*b*c - a*d)*(c + d*x^2)^(5/2))/(5*d^5) + ((
6*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*(c + d*x^2)^(7/2))/(7*d^5) - (2*b*(2*b*c - a*d)*(c + d*x^2)^(9/2))/(9*d^5) +
(b^2*(c + d*x^2)^(11/2))/(11*d^5)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^5 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b x)^2 \sqrt {c+d x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c^2 (b c-a d)^2 \sqrt {c+d x}}{d^4}+\frac {2 c (b c-a d) (-2 b c+a d) (c+d x)^{3/2}}{d^4}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) (c+d x)^{5/2}}{d^4}-\frac {2 b (2 b c-a d) (c+d x)^{7/2}}{d^4}+\frac {b^2 (c+d x)^{9/2}}{d^4}\right ) \, dx,x,x^2\right )\\ &=\frac {c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}{3 d^5}-\frac {2 c (b c-a d) (2 b c-a d) \left (c+d x^2\right )^{5/2}}{5 d^5}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) \left (c+d x^2\right )^{7/2}}{7 d^5}-\frac {2 b (2 b c-a d) \left (c+d x^2\right )^{9/2}}{9 d^5}+\frac {b^2 \left (c+d x^2\right )^{11/2}}{11 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 132, normalized size = 0.84 \begin {gather*} \frac {\left (c+d x^2\right )^{3/2} \left (33 a^2 d^2 \left (8 c^2-12 c d x^2+15 d^2 x^4\right )+22 a b d \left (-16 c^3+24 c^2 d x^2-30 c d^2 x^4+35 d^3 x^6\right )+b^2 \left (128 c^4-192 c^3 d x^2+240 c^2 d^2 x^4-280 c d^3 x^6+315 d^4 x^8\right )\right )}{3465 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2)^2*Sqrt[c + d*x^2],x]

[Out]

((c + d*x^2)^(3/2)*(33*a^2*d^2*(8*c^2 - 12*c*d*x^2 + 15*d^2*x^4) + 22*a*b*d*(-16*c^3 + 24*c^2*d*x^2 - 30*c*d^2
*x^4 + 35*d^3*x^6) + b^2*(128*c^4 - 192*c^3*d*x^2 + 240*c^2*d^2*x^4 - 280*c*d^3*x^6 + 315*d^4*x^8)))/(3465*d^5
)

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IntegrateAlgebraic [A]  time = 0.07, size = 152, normalized size = 0.97 \begin {gather*} \frac {\left (c+d x^2\right )^{3/2} \left (264 a^2 c^2 d^2-396 a^2 c d^3 x^2+495 a^2 d^4 x^4-352 a b c^3 d+528 a b c^2 d^2 x^2-660 a b c d^3 x^4+770 a b d^4 x^6+128 b^2 c^4-192 b^2 c^3 d x^2+240 b^2 c^2 d^2 x^4-280 b^2 c d^3 x^6+315 b^2 d^4 x^8\right )}{3465 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*(a + b*x^2)^2*Sqrt[c + d*x^2],x]

[Out]

((c + d*x^2)^(3/2)*(128*b^2*c^4 - 352*a*b*c^3*d + 264*a^2*c^2*d^2 - 192*b^2*c^3*d*x^2 + 528*a*b*c^2*d^2*x^2 -
396*a^2*c*d^3*x^2 + 240*b^2*c^2*d^2*x^4 - 660*a*b*c*d^3*x^4 + 495*a^2*d^4*x^4 - 280*b^2*c*d^3*x^6 + 770*a*b*d^
4*x^6 + 315*b^2*d^4*x^8))/(3465*d^5)

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fricas [A]  time = 1.53, size = 179, normalized size = 1.14 \begin {gather*} \frac {{\left (315 \, b^{2} d^{5} x^{10} + 35 \, {\left (b^{2} c d^{4} + 22 \, a b d^{5}\right )} x^{8} + 128 \, b^{2} c^{5} - 352 \, a b c^{4} d + 264 \, a^{2} c^{3} d^{2} - 5 \, {\left (8 \, b^{2} c^{2} d^{3} - 22 \, a b c d^{4} - 99 \, a^{2} d^{5}\right )} x^{6} + 3 \, {\left (16 \, b^{2} c^{3} d^{2} - 44 \, a b c^{2} d^{3} + 33 \, a^{2} c d^{4}\right )} x^{4} - 4 \, {\left (16 \, b^{2} c^{4} d - 44 \, a b c^{3} d^{2} + 33 \, a^{2} c^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3465 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

1/3465*(315*b^2*d^5*x^10 + 35*(b^2*c*d^4 + 22*a*b*d^5)*x^8 + 128*b^2*c^5 - 352*a*b*c^4*d + 264*a^2*c^3*d^2 - 5
*(8*b^2*c^2*d^3 - 22*a*b*c*d^4 - 99*a^2*d^5)*x^6 + 3*(16*b^2*c^3*d^2 - 44*a*b*c^2*d^3 + 33*a^2*c*d^4)*x^4 - 4*
(16*b^2*c^4*d - 44*a*b*c^3*d^2 + 33*a^2*c^2*d^3)*x^2)*sqrt(d*x^2 + c)/d^5

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giac [A]  time = 0.29, size = 204, normalized size = 1.30 \begin {gather*} \frac {315 \, {\left (d x^{2} + c\right )}^{\frac {11}{2}} b^{2} - 1540 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} b^{2} c + 2970 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} - 2772 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{3} + 1155 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4} + 770 \, {\left (d x^{2} + c\right )}^{\frac {9}{2}} a b d - 2970 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c d + 4158 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{2} d - 2310 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3} d + 495 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2} - 1386 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c d^{2} + 1155 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2} d^{2}}{3465 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

1/3465*(315*(d*x^2 + c)^(11/2)*b^2 - 1540*(d*x^2 + c)^(9/2)*b^2*c + 2970*(d*x^2 + c)^(7/2)*b^2*c^2 - 2772*(d*x
^2 + c)^(5/2)*b^2*c^3 + 1155*(d*x^2 + c)^(3/2)*b^2*c^4 + 770*(d*x^2 + c)^(9/2)*a*b*d - 2970*(d*x^2 + c)^(7/2)*
a*b*c*d + 4158*(d*x^2 + c)^(5/2)*a*b*c^2*d - 2310*(d*x^2 + c)^(3/2)*a*b*c^3*d + 495*(d*x^2 + c)^(7/2)*a^2*d^2
- 1386*(d*x^2 + c)^(5/2)*a^2*c*d^2 + 1155*(d*x^2 + c)^(3/2)*a^2*c^2*d^2)/d^5

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maple [A]  time = 0.01, size = 149, normalized size = 0.95 \begin {gather*} \frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (315 b^{2} x^{8} d^{4}+770 a b \,d^{4} x^{6}-280 b^{2} c \,d^{3} x^{6}+495 a^{2} d^{4} x^{4}-660 a b c \,d^{3} x^{4}+240 b^{2} c^{2} d^{2} x^{4}-396 a^{2} c \,d^{3} x^{2}+528 a b \,c^{2} d^{2} x^{2}-192 b^{2} c^{3} d \,x^{2}+264 a^{2} c^{2} d^{2}-352 a b \,c^{3} d +128 b^{2} c^{4}\right )}{3465 d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^2+a)^2*(d*x^2+c)^(1/2),x)

[Out]

1/3465*(d*x^2+c)^(3/2)*(315*b^2*d^4*x^8+770*a*b*d^4*x^6-280*b^2*c*d^3*x^6+495*a^2*d^4*x^4-660*a*b*c*d^3*x^4+24
0*b^2*c^2*d^2*x^4-396*a^2*c*d^3*x^2+528*a*b*c^2*d^2*x^2-192*b^2*c^3*d*x^2+264*a^2*c^2*d^2-352*a*b*c^3*d+128*b^
2*c^4)/d^5

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maxima [A]  time = 1.08, size = 249, normalized size = 1.59 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} x^{8}}{11 \, d} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c x^{6}}{99 \, d^{2}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b x^{6}}{9 \, d} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{2} x^{4}}{231 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c x^{4}}{21 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} x^{4}}{7 \, d} - \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} x^{2}}{1155 \, d^{4}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} x^{2}}{105 \, d^{3}} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c x^{2}}{35 \, d^{2}} + \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{4}}{3465 \, d^{5}} - \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{3}}{315 \, d^{4}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{2}}{105 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

1/11*(d*x^2 + c)^(3/2)*b^2*x^8/d - 8/99*(d*x^2 + c)^(3/2)*b^2*c*x^6/d^2 + 2/9*(d*x^2 + c)^(3/2)*a*b*x^6/d + 16
/231*(d*x^2 + c)^(3/2)*b^2*c^2*x^4/d^3 - 4/21*(d*x^2 + c)^(3/2)*a*b*c*x^4/d^2 + 1/7*(d*x^2 + c)^(3/2)*a^2*x^4/
d - 64/1155*(d*x^2 + c)^(3/2)*b^2*c^3*x^2/d^4 + 16/105*(d*x^2 + c)^(3/2)*a*b*c^2*x^2/d^3 - 4/35*(d*x^2 + c)^(3
/2)*a^2*c*x^2/d^2 + 128/3465*(d*x^2 + c)^(3/2)*b^2*c^4/d^5 - 32/315*(d*x^2 + c)^(3/2)*a*b*c^3/d^4 + 8/105*(d*x
^2 + c)^(3/2)*a^2*c^2/d^3

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mupad [B]  time = 0.76, size = 171, normalized size = 1.09 \begin {gather*} \sqrt {d\,x^2+c}\,\left (\frac {264\,a^2\,c^3\,d^2-352\,a\,b\,c^4\,d+128\,b^2\,c^5}{3465\,d^5}+\frac {b^2\,x^{10}}{11}+\frac {x^6\,\left (495\,a^2\,d^5+110\,a\,b\,c\,d^4-40\,b^2\,c^2\,d^3\right )}{3465\,d^5}+\frac {b\,x^8\,\left (22\,a\,d+b\,c\right )}{99\,d}+\frac {c\,x^4\,\left (33\,a^2\,d^2-44\,a\,b\,c\,d+16\,b^2\,c^2\right )}{1155\,d^3}-\frac {4\,c^2\,x^2\,\left (33\,a^2\,d^2-44\,a\,b\,c\,d+16\,b^2\,c^2\right )}{3465\,d^4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^2)^2*(c + d*x^2)^(1/2),x)

[Out]

(c + d*x^2)^(1/2)*((128*b^2*c^5 + 264*a^2*c^3*d^2 - 352*a*b*c^4*d)/(3465*d^5) + (b^2*x^10)/11 + (x^6*(495*a^2*
d^5 - 40*b^2*c^2*d^3 + 110*a*b*c*d^4))/(3465*d^5) + (b*x^8*(22*a*d + b*c))/(99*d) + (c*x^4*(33*a^2*d^2 + 16*b^
2*c^2 - 44*a*b*c*d))/(1155*d^3) - (4*c^2*x^2*(33*a^2*d^2 + 16*b^2*c^2 - 44*a*b*c*d))/(3465*d^4))

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sympy [A]  time = 5.96, size = 389, normalized size = 2.48 \begin {gather*} \begin {cases} \frac {8 a^{2} c^{3} \sqrt {c + d x^{2}}}{105 d^{3}} - \frac {4 a^{2} c^{2} x^{2} \sqrt {c + d x^{2}}}{105 d^{2}} + \frac {a^{2} c x^{4} \sqrt {c + d x^{2}}}{35 d} + \frac {a^{2} x^{6} \sqrt {c + d x^{2}}}{7} - \frac {32 a b c^{4} \sqrt {c + d x^{2}}}{315 d^{4}} + \frac {16 a b c^{3} x^{2} \sqrt {c + d x^{2}}}{315 d^{3}} - \frac {4 a b c^{2} x^{4} \sqrt {c + d x^{2}}}{105 d^{2}} + \frac {2 a b c x^{6} \sqrt {c + d x^{2}}}{63 d} + \frac {2 a b x^{8} \sqrt {c + d x^{2}}}{9} + \frac {128 b^{2} c^{5} \sqrt {c + d x^{2}}}{3465 d^{5}} - \frac {64 b^{2} c^{4} x^{2} \sqrt {c + d x^{2}}}{3465 d^{4}} + \frac {16 b^{2} c^{3} x^{4} \sqrt {c + d x^{2}}}{1155 d^{3}} - \frac {8 b^{2} c^{2} x^{6} \sqrt {c + d x^{2}}}{693 d^{2}} + \frac {b^{2} c x^{8} \sqrt {c + d x^{2}}}{99 d} + \frac {b^{2} x^{10} \sqrt {c + d x^{2}}}{11} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\frac {a^{2} x^{6}}{6} + \frac {a b x^{8}}{4} + \frac {b^{2} x^{10}}{10}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**2+a)**2*(d*x**2+c)**(1/2),x)

[Out]

Piecewise((8*a**2*c**3*sqrt(c + d*x**2)/(105*d**3) - 4*a**2*c**2*x**2*sqrt(c + d*x**2)/(105*d**2) + a**2*c*x**
4*sqrt(c + d*x**2)/(35*d) + a**2*x**6*sqrt(c + d*x**2)/7 - 32*a*b*c**4*sqrt(c + d*x**2)/(315*d**4) + 16*a*b*c*
*3*x**2*sqrt(c + d*x**2)/(315*d**3) - 4*a*b*c**2*x**4*sqrt(c + d*x**2)/(105*d**2) + 2*a*b*c*x**6*sqrt(c + d*x*
*2)/(63*d) + 2*a*b*x**8*sqrt(c + d*x**2)/9 + 128*b**2*c**5*sqrt(c + d*x**2)/(3465*d**5) - 64*b**2*c**4*x**2*sq
rt(c + d*x**2)/(3465*d**4) + 16*b**2*c**3*x**4*sqrt(c + d*x**2)/(1155*d**3) - 8*b**2*c**2*x**6*sqrt(c + d*x**2
)/(693*d**2) + b**2*c*x**8*sqrt(c + d*x**2)/(99*d) + b**2*x**10*sqrt(c + d*x**2)/11, Ne(d, 0)), (sqrt(c)*(a**2
*x**6/6 + a*b*x**8/4 + b**2*x**10/10), True))

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