3.6.93 \(\int \frac {(a+b x^2)^2 \sqrt {c+d x^2}}{x^{10}} \, dx\)

Optimal. Leaf size=143 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac {2 d \left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{315 c^4 x^3}-\frac {\left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{105 c^3 x^5}-\frac {2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{21 c^2 x^7} \]

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Rubi [A]  time = 0.13, antiderivative size = 144, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 453, 271, 264} \begin {gather*} -\frac {\left (c+d x^2\right )^{3/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right )}{105 c^3 x^5}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac {2 d \left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{315 c^4 x^3}-\frac {2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{21 c^2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]

[Out]

-(a^2*(c + d*x^2)^(3/2))/(9*c*x^9) - (2*a*(3*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2*x^7) - ((21*b^2*c^2 - 24*a*
b*c*d + 8*a^2*d^2)*(c + d*x^2)^(3/2))/(105*c^3*x^5) + (2*d*(21*b^2*c^2 - 8*a*d*(3*b*c - a*d))*(c + d*x^2)^(3/2
))/(315*c^4*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^{10}} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac {\int \frac {\left (6 a (3 b c-a d)+9 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^8} \, dx}{9 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {1}{21} \left (-21 b^2+\frac {8 a d (3 b c-a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^6} \, dx\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {\left (21 b^2-\frac {8 a d (3 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{105 c x^5}-\frac {\left (2 d \left (21 b^2 c^2-24 a b c d+8 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{x^4} \, dx}{105 c^3}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}-\frac {2 a (3 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^7}-\frac {\left (21 b^2-\frac {8 a d (3 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{105 c x^5}+\frac {2 d \left (21 b^2 c^2-24 a b c d+8 a^2 d^2\right ) \left (c+d x^2\right )^{3/2}}{315 c^4 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 108, normalized size = 0.76 \begin {gather*} -\frac {\left (c+d x^2\right )^{3/2} \left (a^2 \left (35 c^3-30 c^2 d x^2+24 c d^2 x^4-16 d^3 x^6\right )+6 a b c x^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+21 b^2 c^2 x^4 \left (3 c-2 d x^2\right )\right )}{315 c^4 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]

[Out]

-1/315*((c + d*x^2)^(3/2)*(21*b^2*c^2*x^4*(3*c - 2*d*x^2) + 6*a*b*c*x^2*(15*c^2 - 12*c*d*x^2 + 8*d^2*x^4) + a^
2*(35*c^3 - 30*c^2*d*x^2 + 24*c*d^2*x^4 - 16*d^3*x^6)))/(c^4*x^9)

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IntegrateAlgebraic [A]  time = 0.24, size = 161, normalized size = 1.13 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-35 a^2 c^4-5 a^2 c^3 d x^2+6 a^2 c^2 d^2 x^4-8 a^2 c d^3 x^6+16 a^2 d^4 x^8-90 a b c^4 x^2-18 a b c^3 d x^4+24 a b c^2 d^2 x^6-48 a b c d^3 x^8-63 b^2 c^4 x^4-21 b^2 c^3 d x^6+42 b^2 c^2 d^2 x^8\right )}{315 c^4 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^10,x]

[Out]

(Sqrt[c + d*x^2]*(-35*a^2*c^4 - 90*a*b*c^4*x^2 - 5*a^2*c^3*d*x^2 - 63*b^2*c^4*x^4 - 18*a*b*c^3*d*x^4 + 6*a^2*c
^2*d^2*x^4 - 21*b^2*c^3*d*x^6 + 24*a*b*c^2*d^2*x^6 - 8*a^2*c*d^3*x^6 + 42*b^2*c^2*d^2*x^8 - 48*a*b*c*d^3*x^8 +
16*a^2*d^4*x^8))/(315*c^4*x^9)

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fricas [A]  time = 2.04, size = 147, normalized size = 1.03 \begin {gather*} \frac {{\left (2 \, {\left (21 \, b^{2} c^{2} d^{2} - 24 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )} x^{8} - {\left (21 \, b^{2} c^{3} d - 24 \, a b c^{2} d^{2} + 8 \, a^{2} c d^{3}\right )} x^{6} - 35 \, a^{2} c^{4} - 3 \, {\left (21 \, b^{2} c^{4} + 6 \, a b c^{3} d - 2 \, a^{2} c^{2} d^{2}\right )} x^{4} - 5 \, {\left (18 \, a b c^{4} + a^{2} c^{3} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{315 \, c^{4} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x, algorithm="fricas")

[Out]

1/315*(2*(21*b^2*c^2*d^2 - 24*a*b*c*d^3 + 8*a^2*d^4)*x^8 - (21*b^2*c^3*d - 24*a*b*c^2*d^2 + 8*a^2*c*d^3)*x^6 -
35*a^2*c^4 - 3*(21*b^2*c^4 + 6*a*b*c^3*d - 2*a^2*c^2*d^2)*x^4 - 5*(18*a*b*c^4 + a^2*c^3*d)*x^2)*sqrt(d*x^2 +
c)/(c^4*x^9)

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giac [B]  time = 0.44, size = 579, normalized size = 4.05 \begin {gather*} \frac {4 \, {\left (315 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{14} b^{2} d^{\frac {5}{2}} - 1155 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} b^{2} c d^{\frac {5}{2}} + 1680 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{12} a b d^{\frac {7}{2}} + 1575 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{2} c^{2} d^{\frac {5}{2}} - 2520 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b c d^{\frac {7}{2}} + 2520 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a^{2} d^{\frac {9}{2}} - 1071 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{3} d^{\frac {5}{2}} + 504 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c^{2} d^{\frac {7}{2}} + 1512 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} c d^{\frac {9}{2}} + 609 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{4} d^{\frac {5}{2}} - 336 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{3} d^{\frac {7}{2}} + 672 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} c^{2} d^{\frac {9}{2}} - 441 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{5} d^{\frac {5}{2}} + 864 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{4} d^{\frac {7}{2}} - 288 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{3} d^{\frac {9}{2}} + 189 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{6} d^{\frac {5}{2}} - 216 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{5} d^{\frac {7}{2}} + 72 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{4} d^{\frac {9}{2}} - 21 \, b^{2} c^{7} d^{\frac {5}{2}} + 24 \, a b c^{6} d^{\frac {7}{2}} - 8 \, a^{2} c^{5} d^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x, algorithm="giac")

[Out]

4/315*(315*(sqrt(d)*x - sqrt(d*x^2 + c))^14*b^2*d^(5/2) - 1155*(sqrt(d)*x - sqrt(d*x^2 + c))^12*b^2*c*d^(5/2)
+ 1680*(sqrt(d)*x - sqrt(d*x^2 + c))^12*a*b*d^(7/2) + 1575*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^2*c^2*d^(5/2) -
2520*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b*c*d^(7/2) + 2520*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a^2*d^(9/2) - 1071
*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^3*d^(5/2) + 504*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b*c^2*d^(7/2) + 1512*
(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*c*d^(9/2) + 609*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^4*d^(5/2) - 336*(sqr
t(d)*x - sqrt(d*x^2 + c))^6*a*b*c^3*d^(7/2) + 672*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*c^2*d^(9/2) - 441*(sqrt(
d)*x - sqrt(d*x^2 + c))^4*b^2*c^5*d^(5/2) + 864*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^4*d^(7/2) - 288*(sqrt(d)
*x - sqrt(d*x^2 + c))^4*a^2*c^3*d^(9/2) + 189*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^6*d^(5/2) - 216*(sqrt(d)*x
- sqrt(d*x^2 + c))^2*a*b*c^5*d^(7/2) + 72*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^4*d^(9/2) - 21*b^2*c^7*d^(5/2
) + 24*a*b*c^6*d^(7/2) - 8*a^2*c^5*d^(9/2))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^9

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maple [A]  time = 0.01, size = 117, normalized size = 0.82 \begin {gather*} -\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (-16 a^{2} d^{3} x^{6}+48 a b c \,d^{2} x^{6}-42 b^{2} c^{2} d \,x^{6}+24 a^{2} c \,d^{2} x^{4}-72 a b \,c^{2} d \,x^{4}+63 b^{2} c^{3} x^{4}-30 a^{2} c^{2} d \,x^{2}+90 a b \,c^{3} x^{2}+35 a^{2} c^{3}\right )}{315 c^{4} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x)

[Out]

-1/315*(d*x^2+c)^(3/2)*(-16*a^2*d^3*x^6+48*a*b*c*d^2*x^6-42*b^2*c^2*d*x^6+24*a^2*c*d^2*x^4-72*a*b*c^2*d*x^4+63
*b^2*c^3*x^4-30*a^2*c^2*d*x^2+90*a*b*c^3*x^2+35*a^2*c^3)/x^9/c^4

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maxima [A]  time = 1.16, size = 190, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d}{15 \, c^{2} x^{3}} - \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{105 \, c^{3} x^{3}} + \frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{315 \, c^{4} x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{5 \, c x^{5}} + \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{35 \, c^{2} x^{5}} - \frac {8 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2}}{105 \, c^{3} x^{5}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{7 \, c x^{7}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{21 \, c^{2} x^{7}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{9 \, c x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^10,x, algorithm="maxima")

[Out]

2/15*(d*x^2 + c)^(3/2)*b^2*d/(c^2*x^3) - 16/105*(d*x^2 + c)^(3/2)*a*b*d^2/(c^3*x^3) + 16/315*(d*x^2 + c)^(3/2)
*a^2*d^3/(c^4*x^3) - 1/5*(d*x^2 + c)^(3/2)*b^2/(c*x^5) + 8/35*(d*x^2 + c)^(3/2)*a*b*d/(c^2*x^5) - 8/105*(d*x^2
+ c)^(3/2)*a^2*d^2/(c^3*x^5) - 2/7*(d*x^2 + c)^(3/2)*a*b/(c*x^7) + 2/21*(d*x^2 + c)^(3/2)*a^2*d/(c^2*x^7) - 1
/9*(d*x^2 + c)^(3/2)*a^2/(c*x^9)

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mupad [B]  time = 2.29, size = 249, normalized size = 1.74 \begin {gather*} \frac {2\,a^2\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^5}-\frac {b^2\,\sqrt {d\,x^2+c}}{5\,x^5}-\frac {2\,a\,b\,\sqrt {d\,x^2+c}}{7\,x^7}-\frac {a^2\,\sqrt {d\,x^2+c}}{9\,x^9}-\frac {8\,a^2\,d^3\,\sqrt {d\,x^2+c}}{315\,c^3\,x^3}+\frac {16\,a^2\,d^4\,\sqrt {d\,x^2+c}}{315\,c^4\,x}+\frac {2\,b^2\,d^2\,\sqrt {d\,x^2+c}}{15\,c^2\,x}-\frac {a^2\,d\,\sqrt {d\,x^2+c}}{63\,c\,x^7}-\frac {b^2\,d\,\sqrt {d\,x^2+c}}{15\,c\,x^3}+\frac {8\,a\,b\,d^2\,\sqrt {d\,x^2+c}}{105\,c^2\,x^3}-\frac {16\,a\,b\,d^3\,\sqrt {d\,x^2+c}}{105\,c^3\,x}-\frac {2\,a\,b\,d\,\sqrt {d\,x^2+c}}{35\,c\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^10,x)

[Out]

(2*a^2*d^2*(c + d*x^2)^(1/2))/(105*c^2*x^5) - (b^2*(c + d*x^2)^(1/2))/(5*x^5) - (2*a*b*(c + d*x^2)^(1/2))/(7*x
^7) - (a^2*(c + d*x^2)^(1/2))/(9*x^9) - (8*a^2*d^3*(c + d*x^2)^(1/2))/(315*c^3*x^3) + (16*a^2*d^4*(c + d*x^2)^
(1/2))/(315*c^4*x) + (2*b^2*d^2*(c + d*x^2)^(1/2))/(15*c^2*x) - (a^2*d*(c + d*x^2)^(1/2))/(63*c*x^7) - (b^2*d*
(c + d*x^2)^(1/2))/(15*c*x^3) + (8*a*b*d^2*(c + d*x^2)^(1/2))/(105*c^2*x^3) - (16*a*b*d^3*(c + d*x^2)^(1/2))/(
105*c^3*x) - (2*a*b*d*(c + d*x^2)^(1/2))/(35*c*x^5)

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sympy [B]  time = 5.36, size = 1061, normalized size = 7.42 \begin {gather*} - \frac {35 a^{2} c^{7} d^{\frac {19}{2}} \sqrt {\frac {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}} - \frac {110 a^{2} c^{6} d^{\frac {21}{2}} x^{2} \sqrt {\frac {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}} - \frac {114 a^{2} c^{5} d^{\frac {23}{2}} x^{4} \sqrt {\frac {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}} - \frac {40 a^{2} c^{4} d^{\frac {25}{2}} x^{6} \sqrt {\frac {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}} + \frac {5 a^{2} c^{3} d^{\frac {27}{2}} x^{8} \sqrt {\frac {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}} + \frac {30 a^{2} c^{2} d^{\frac {29}{2}} x^{10} \sqrt {\frac {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}} + \frac {40 a^{2} c d^{\frac {31}{2}} x^{12} \sqrt {\frac {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}} + \frac {16 a^{2} d^{\frac {33}{2}} x^{14} \sqrt {\frac {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}} - \frac {30 a b c^{5} d^{\frac {9}{2}} \sqrt {\frac {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}} - \frac {66 a b c^{4} d^{\frac {11}{2}} x^{2} \sqrt {\frac {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}} - \frac {34 a b c^{3} d^{\frac {13}{2}} x^{4} \sqrt {\frac {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}} - \frac {6 a b c^{2} d^{\frac {15}{2}} x^{6} \sqrt {\frac {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}} - \frac {24 a b c d^{\frac {17}{2}} x^{8} \sqrt {\frac {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}} - \frac {16 a b d^{\frac {19}{2}} x^{10} \sqrt {\frac {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}} - \frac {b^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {b^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c x^{2}} + \frac {2 b^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**10,x)

[Out]

-35*a**2*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) - 110*a**2*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) - 114*a**2*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) - 40*a**2*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) + 5*a**2*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 + 315*c**4*d**12*x**14) + 30*a**2*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) + 40*a**2*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) + 16*a**2
*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) - 30*a*b*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) - 66*a*b*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 + 1
05*c**3*d**6*x**10) - 34*a*b*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) - 6*a*b*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) - 24*a*b*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) - 16*a*b*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) - b**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(5*x**4) - b**2*d**(3/2)*sqrt(c/(d*x**2) + 1)/
(15*c*x**2) + 2*b**2*d**(5/2)*sqrt(c/(d*x**2) + 1)/(15*c**2)

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