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

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

[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 - 8*a*d
*(3*b*c - a*d))*(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)

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Rubi [A]  time = 0.131767, 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} \[ -\frac{\left (c+d x^2\right )^{3/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right )}{105 c^3 x^5}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{9 c x^9}+\frac{2 d \left (c+d x^2\right )^{3/2} \left (21 b^2 c^2-8 a d (3 b c-a d)\right )}{315 c^4 x^3}-\frac{2 a \left (c+d x^2\right )^{3/2} (3 b c-a d)}{21 c^2 x^7} \]

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 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]

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 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 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]

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*}

Mathematica [A]  time = 0.0661442, size = 108, normalized size = 0.76 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (a^2 \left (-30 c^2 d x^2+35 c^3+24 c d^2 x^4-16 d^3 x^6\right )+6 a b c x^2 \left (15 c^2-12 c d x^2+8 d^2 x^4\right )+21 b^2 c^2 x^4 \left (3 c-2 d x^2\right )\right )}{315 c^4 x^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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)))/(315*c^4*x^9)

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Maple [A]  time = 0.007, size = 117, normalized size = 0.8 \begin{align*} -{\frac{-16\,{x}^{6}{a}^{2}{d}^{3}+48\,{x}^{6}abc{d}^{2}-42\,{x}^{6}{b}^{2}{c}^{2}d+24\,{x}^{4}{a}^{2}c{d}^{2}-72\,{x}^{4}ab{c}^{2}d+63\,{x}^{4}{b}^{2}{c}^{3}-30\,{x}^{2}{a}^{2}{c}^{2}d+90\,{x}^{2}ab{c}^{3}+35\,{a}^{2}{c}^{3}}{315\,{x}^{9}{c}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.58041, size = 316, normalized size = 2.21 \begin{align*} \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{align*}

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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Sympy [B]  time = 4.86183, size = 1061, normalized size = 7.42 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [B]  time = 1.15167, size = 782, normalized size = 5.47 \begin{align*} \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{align*}

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