3.612 \(\int \frac {(a+b x^2)^2 \sqrt {c+d x^2}}{x^{12}} \, dx\)

Optimal. Leaf size=189 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {8 d^2 \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{3465 c^5 x^3}+\frac {4 d \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{1155 c^4 x^5}-\frac {\left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{231 c^3 x^7}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{99 c^2 x^9} \]

[Out]

-1/11*a^2*(d*x^2+c)^(3/2)/c/x^11-2/99*a*(-4*a*d+11*b*c)*(d*x^2+c)^(3/2)/c^2/x^9-1/231*(33*b^2*c^2-4*a*d*(-4*a*
d+11*b*c))*(d*x^2+c)^(3/2)/c^3/x^7+4/1155*d*(33*b^2*c^2-4*a*d*(-4*a*d+11*b*c))*(d*x^2+c)^(3/2)/c^4/x^5-8/3465*
d^2*(33*b^2*c^2-4*a*d*(-4*a*d+11*b*c))*(d*x^2+c)^(3/2)/c^5/x^3

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Rubi [A]  time = 0.17, antiderivative size = 190, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 453, 271, 264} \[ -\frac {\left (c+d x^2\right )^{3/2} \left (16 a^2 d^2-44 a b c d+33 b^2 c^2\right )}{231 c^3 x^7}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {8 d^2 \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{3465 c^5 x^3}+\frac {4 d \left (c+d x^2\right )^{3/2} \left (33 b^2 c^2-4 a d (11 b c-4 a d)\right )}{1155 c^4 x^5}-\frac {2 a \left (c+d x^2\right )^{3/2} (11 b c-4 a d)}{99 c^2 x^9} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^12,x]

[Out]

-(a^2*(c + d*x^2)^(3/2))/(11*c*x^11) - (2*a*(11*b*c - 4*a*d)*(c + d*x^2)^(3/2))/(99*c^2*x^9) - ((33*b^2*c^2 -
44*a*b*c*d + 16*a^2*d^2)*(c + d*x^2)^(3/2))/(231*c^3*x^7) + (4*d*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*
x^2)^(3/2))/(1155*c^4*x^5) - (8*d^2*(33*b^2*c^2 - 4*a*d*(11*b*c - 4*a*d))*(c + d*x^2)^(3/2))/(3465*c^5*x^3)

Rule 264

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

Rule 271

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

Rule 453

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

Rule 462

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(c^2*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{12}} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}+\frac {\int \frac {\left (2 a (11 b c-4 a d)+11 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^{10}} \, dx}{11 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {1}{33} \left (-33 b^2+\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^8} \, dx\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}-\frac {\left (4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^6} \, dx}{231 c^3}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}+\frac {4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}+\frac {\left (8 d^2 \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^4} \, dx}{1155 c^4}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11}}-\frac {2 a (11 b c-4 a d) \left (c+d x^2\right )^{3/2}}{99 c^2 x^9}-\frac {\left (33 b^2-\frac {4 a d (11 b c-4 a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{231 c x^7}+\frac {4 d \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{1155 c^4 x^5}-\frac {8 d^2 \left (33 b^2 c^2-44 a b c d+16 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{3465 c^5 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 141, normalized size = 0.75 \[ -\frac {\left (c+d x^2\right )^{3/2} \left (a^2 \left (315 c^4-280 c^3 d x^2+240 c^2 d^2 x^4-192 c d^3 x^6+128 d^4 x^8\right )+22 a b c x^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )+33 b^2 c^2 x^4 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )\right )}{3465 c^5 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^12,x]

[Out]

-1/3465*((c + d*x^2)^(3/2)*(33*b^2*c^2*x^4*(15*c^2 - 12*c*d*x^2 + 8*d^2*x^4) + 22*a*b*c*x^2*(35*c^3 - 30*c^2*d
*x^2 + 24*c*d^2*x^4 - 16*d^3*x^6) + a^2*(315*c^4 - 280*c^3*d*x^2 + 240*c^2*d^2*x^4 - 192*c*d^3*x^6 + 128*d^4*x
^8)))/(c^5*x^11)

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fricas [A]  time = 1.00, size = 185, normalized size = 0.98 \[ -\frac {{\left (8 \, {\left (33 \, b^{2} c^{2} d^{3} - 44 \, a b c d^{4} + 16 \, a^{2} d^{5}\right )} x^{10} - 4 \, {\left (33 \, b^{2} c^{3} d^{2} - 44 \, a b c^{2} d^{3} + 16 \, a^{2} c d^{4}\right )} x^{8} + 315 \, a^{2} c^{5} + 3 \, {\left (33 \, b^{2} c^{4} d - 44 \, a b c^{3} d^{2} + 16 \, a^{2} c^{2} d^{3}\right )} x^{6} + 5 \, {\left (99 \, b^{2} c^{5} + 22 \, a b c^{4} d - 8 \, a^{2} c^{3} d^{2}\right )} x^{4} + 35 \, {\left (22 \, a b c^{5} + a^{2} c^{4} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{3465 \, c^{5} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x, algorithm="fricas")

[Out]

-1/3465*(8*(33*b^2*c^2*d^3 - 44*a*b*c*d^4 + 16*a^2*d^5)*x^10 - 4*(33*b^2*c^3*d^2 - 44*a*b*c^2*d^3 + 16*a^2*c*d
^4)*x^8 + 315*a^2*c^5 + 3*(33*b^2*c^4*d - 44*a*b*c^3*d^2 + 16*a^2*c^2*d^3)*x^6 + 5*(99*b^2*c^5 + 22*a*b*c^4*d
- 8*a^2*c^3*d^2)*x^4 + 35*(22*a*b*c^5 + a^2*c^4*d)*x^2)*sqrt(d*x^2 + c)/(c^5*x^11)

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giac [B]  time = 0.42, size = 668, normalized size = 3.53 \[ \frac {16 \, {\left (2310 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{16} b^{2} d^{\frac {7}{2}} - 8085 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} b^{2} c d^{\frac {7}{2}} + 13860 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} a b d^{\frac {9}{2}} + 9933 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} c^{2} d^{\frac {7}{2}} - 19404 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a b c d^{\frac {9}{2}} + 22176 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a^{2} d^{\frac {11}{2}} - 5313 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c^{3} d^{\frac {7}{2}} + 924 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b c^{2} d^{\frac {9}{2}} + 14784 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a^{2} c d^{\frac {11}{2}} + 2805 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{4} d^{\frac {7}{2}} - 660 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{3} d^{\frac {9}{2}} + 5280 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c^{2} d^{\frac {11}{2}} - 3135 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{5} d^{\frac {7}{2}} + 7260 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{4} d^{\frac {9}{2}} - 2640 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{3} d^{\frac {11}{2}} + 1815 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{6} d^{\frac {7}{2}} - 2420 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{5} d^{\frac {9}{2}} + 880 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{4} d^{\frac {11}{2}} - 363 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{7} d^{\frac {7}{2}} + 484 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{6} d^{\frac {9}{2}} - 176 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{5} d^{\frac {11}{2}} + 33 \, b^{2} c^{8} d^{\frac {7}{2}} - 44 \, a b c^{7} d^{\frac {9}{2}} + 16 \, a^{2} c^{6} d^{\frac {11}{2}}\right )}}{3465 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x, algorithm="giac")

[Out]

16/3465*(2310*(sqrt(d)*x - sqrt(d*x^2 + c))^16*b^2*d^(7/2) - 8085*(sqrt(d)*x - sqrt(d*x^2 + c))^14*b^2*c*d^(7/
2) + 13860*(sqrt(d)*x - sqrt(d*x^2 + c))^14*a*b*d^(9/2) + 9933*(sqrt(d)*x - sqrt(d*x^2 + c))^12*b^2*c^2*d^(7/2
) - 19404*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a*b*c*d^(9/2) + 22176*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a^2*d^(11/2)
- 5313*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^3*d^(7/2) + 924*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*c^2*d^(9/2
) + 14784*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a^2*c*d^(11/2) + 2805*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^4*d^(7/
2) - 660*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c^3*d^(9/2) + 5280*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*c^2*d^(11/
2) - 3135*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^5*d^(7/2) + 7260*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c^4*d^(9/
2) - 2640*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c^3*d^(11/2) + 1815*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^6*d^(7
/2) - 2420*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^5*d^(9/2) + 880*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c^4*d^(11
/2) - 363*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^7*d^(7/2) + 484*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^6*d^(9/2
) - 176*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^5*d^(11/2) + 33*b^2*c^8*d^(7/2) - 44*a*b*c^7*d^(9/2) + 16*a^2*c^
6*d^(11/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^11

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maple [A]  time = 0.01, size = 158, normalized size = 0.84 \[ -\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (128 a^{2} d^{4} x^{8}-352 a b c \,d^{3} x^{8}+264 b^{2} c^{2} d^{2} x^{8}-192 a^{2} c \,d^{3} x^{6}+528 a b \,c^{2} d^{2} x^{6}-396 b^{2} c^{3} d \,x^{6}+240 a^{2} c^{2} d^{2} x^{4}-660 a b \,c^{3} d \,x^{4}+495 b^{2} c^{4} x^{4}-280 a^{2} c^{3} d \,x^{2}+770 a b \,c^{4} x^{2}+315 a^{2} c^{4}\right )}{3465 c^{5} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x)

[Out]

-1/3465*(d*x^2+c)^(3/2)*(128*a^2*d^4*x^8-352*a*b*c*d^3*x^8+264*b^2*c^2*d^2*x^8-192*a^2*c*d^3*x^6+528*a*b*c^2*d
^2*x^6-396*b^2*c^3*d*x^6+240*a^2*c^2*d^2*x^4-660*a*b*c^3*d*x^4+495*b^2*c^4*x^4-280*a^2*c^3*d*x^2+770*a*b*c^4*x
^2+315*a^2*c^4)/x^11/c^5

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maxima [A]  time = 1.12, size = 258, normalized size = 1.37 \[ -\frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{2}}{105 \, c^{3} x^{3}} + \frac {32 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{3}}{315 \, c^{4} x^{3}} - \frac {128 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{4}}{3465 \, c^{5} x^{3}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{35 \, c^{2} x^{5}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{105 \, c^{3} x^{5}} + \frac {64 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{1155 \, c^{4} x^{5}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{7 \, c x^{7}} + \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{21 \, c^{2} x^{7}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{231 \, c^{3} x^{7}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{9 \, c x^{9}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{99 \, c^{2} x^{9}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{11 \, c x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^12,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.99, size = 317, normalized size = 1.68 \[ \frac {8\,a^2\,d^2\,\sqrt {d\,x^2+c}}{693\,c^2\,x^7}-\frac {b^2\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {a^2\,\sqrt {d\,x^2+c}}{11\,x^{11}}-\frac {16\,a^2\,d^3\,\sqrt {d\,x^2+c}}{1155\,c^3\,x^5}+\frac {64\,a^2\,d^4\,\sqrt {d\,x^2+c}}{3465\,c^4\,x^3}-\frac {128\,a^2\,d^5\,\sqrt {d\,x^2+c}}{3465\,c^5\,x}+\frac {4\,b^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {8\,b^2\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{99\,c\,x^9}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5}+\frac {4\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^5}-\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{315\,c^3\,x^3}+\frac {32\,a\,b\,d^4\,\sqrt {d\,x^2+c}}{315\,c^4\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{63\,c\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^12,x)

[Out]

(8*a^2*d^2*(c + d*x^2)^(1/2))/(693*c^2*x^7) - (b^2*(c + d*x^2)^(1/2))/(7*x^7) - (2*a*b*(c + d*x^2)^(1/2))/(9*x
^9) - (a^2*(c + d*x^2)^(1/2))/(11*x^11) - (16*a^2*d^3*(c + d*x^2)^(1/2))/(1155*c^3*x^5) + (64*a^2*d^4*(c + d*x
^2)^(1/2))/(3465*c^4*x^3) - (128*a^2*d^5*(c + d*x^2)^(1/2))/(3465*c^5*x) + (4*b^2*d^2*(c + d*x^2)^(1/2))/(105*
c^2*x^3) - (8*b^2*d^3*(c + d*x^2)^(1/2))/(105*c^3*x) - (a^2*d*(c + d*x^2)^(1/2))/(99*c*x^9) - (b^2*d*(c + d*x^
2)^(1/2))/(35*c*x^5) + (4*a*b*d^2*(c + d*x^2)^(1/2))/(105*c^2*x^5) - (16*a*b*d^3*(c + d*x^2)^(1/2))/(315*c^3*x
^3) + (32*a*b*d^4*(c + d*x^2)^(1/2))/(315*c^4*x) - (2*a*b*d*(c + d*x^2)^(1/2))/(63*c*x^7)

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sympy [B]  time = 8.28, size = 1856, normalized size = 9.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**12,x)

[Out]

-315*a**2*c**9*d**(33/2)*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**
18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 1295*a**2*c**8*d**(35/2)*x**2*sqrt(c/(d*x**2) + 1
)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**
5*d**20*x**18) - 1990*a**2*c**7*d**(37/2)*x**4*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*
x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 1358*a**2*c**6*d**(39/2)*x*
*6*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*
d**19*x**16 + 3465*c**5*d**20*x**18) - 343*a**2*c**5*d**(41/2)*x**8*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**1
0 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 35*a**
2*c**4*d**(43/2)*x**10*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18
*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 280*a**2*c**3*d**(45/2)*x**12*sqrt(c/(d*x**2) + 1)/
(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*
d**20*x**18) - 560*a**2*c**2*d**(47/2)*x**14*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x*
*12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 448*a**2*c*d**(49/2)*x**16*sq
rt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19
*x**16 + 3465*c**5*d**20*x**18) - 128*a**2*d**(51/2)*x**18*sqrt(c/(d*x**2) + 1)/(3465*c**9*d**16*x**10 + 13860
*c**8*d**17*x**12 + 20790*c**7*d**18*x**14 + 13860*c**6*d**19*x**16 + 3465*c**5*d**20*x**18) - 70*a*b*c**7*d**
(19/2)*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12
*x**14) - 220*a*b*c**6*d**(21/2)*x**2*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c*
*5*d**11*x**12 + 315*c**4*d**12*x**14) - 228*a*b*c**5*d**(23/2)*x**4*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8
+ 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) - 80*a*b*c**4*d**(25/2)*x**6*sqrt(c/(d*x
**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) + 10*a*b*c
**3*d**(27/2)*x**8*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 31
5*c**4*d**12*x**14) + 60*a*b*c**2*d**(29/2)*x**10*sqrt(c/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x*
*10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) + 80*a*b*c*d**(31/2)*x**12*sqrt(c/(d*x**2) + 1)/(315*c**7*d
**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) + 32*a*b*d**(33/2)*x**14*sqrt(c
/(d*x**2) + 1)/(315*c**7*d**9*x**8 + 945*c**6*d**10*x**10 + 945*c**5*d**11*x**12 + 315*c**4*d**12*x**14) - 15*
b**2*c**5*d**(9/2)*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) - 33*b
**2*c**4*d**(11/2)*x**2*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**10) -
17*b**2*c**3*d**(13/2)*x**4*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6*x**
10) - 3*b**2*c**2*d**(15/2)*x**6*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d**6
*x**10) - 12*b**2*c*d**(17/2)*x**8*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d*
*6*x**10) - 8*b**2*d**(19/2)*x**10*sqrt(c/(d*x**2) + 1)/(105*c**5*d**4*x**6 + 210*c**4*d**5*x**8 + 105*c**3*d*
*6*x**10)

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