3.553 \(\int \frac{(a+b x^2)^{5/2} (A+B x^2)}{x^{10}} \, dx\)

Optimal. Leaf size=53 \[ \frac{\left (a+b x^2\right )^{7/2} (2 A b-9 a B)}{63 a^2 x^7}-\frac{A \left (a+b x^2\right )^{7/2}}{9 a x^9} \]

[Out]

-(A*(a + b*x^2)^(7/2))/(9*a*x^9) + ((2*A*b - 9*a*B)*(a + b*x^2)^(7/2))/(63*a^2*x^7)

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Rubi [A]  time = 0.0215579, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 264} \[ \frac{\left (a+b x^2\right )^{7/2} (2 A b-9 a B)}{63 a^2 x^7}-\frac{A \left (a+b x^2\right )^{7/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^10,x]

[Out]

-(A*(a + b*x^2)^(7/2))/(9*a*x^9) + ((2*A*b - 9*a*B)*(a + b*x^2)^(7/2))/(63*a^2*x^7)

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 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 )^{5/2} \left (A+B x^2\right )}{x^{10}} \, dx &=-\frac{A \left (a+b x^2\right )^{7/2}}{9 a x^9}-\frac{(2 A b-9 a B) \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{9 a}\\ &=-\frac{A \left (a+b x^2\right )^{7/2}}{9 a x^9}+\frac{(2 A b-9 a B) \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}\\ \end{align*}

Mathematica [A]  time = 0.017225, size = 40, normalized size = 0.75 \[ -\frac{\left (a+b x^2\right )^{7/2} \left (7 a A+9 a B x^2-2 A b x^2\right )}{63 a^2 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^10,x]

[Out]

-((a + b*x^2)^(7/2)*(7*a*A - 2*A*b*x^2 + 9*a*B*x^2))/(63*a^2*x^9)

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Maple [A]  time = 0.004, size = 37, normalized size = 0.7 \begin{align*} -{\frac{-2\,Ab{x}^{2}+9\,Ba{x}^{2}+7\,Aa}{63\,{x}^{9}{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)*(B*x^2+A)/x^10,x)

[Out]

-1/63*(b*x^2+a)^(7/2)*(-2*A*b*x^2+9*B*a*x^2+7*A*a)/x^9/a^2

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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)^(5/2)*(B*x^2+A)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.86283, size = 223, normalized size = 4.21 \begin{align*} -\frac{{\left ({\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{8} +{\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{6} + 7 \, A a^{4} + 3 \,{\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{4} +{\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{63 \, a^{2} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(B*x^2+A)/x^10,x, algorithm="fricas")

[Out]

-1/63*((9*B*a*b^3 - 2*A*b^4)*x^8 + (27*B*a^2*b^2 + A*a*b^3)*x^6 + 7*A*a^4 + 3*(9*B*a^3*b + 5*A*a^2*b^2)*x^4 +
(9*B*a^4 + 19*A*a^3*b)*x^2)*sqrt(b*x^2 + a)/(a^2*x^9)

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Sympy [B]  time = 8.25986, size = 1489, normalized size = 28.09 \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)**(5/2)*(B*x**2+A)/x**10,x)

[Out]

-35*A*a**9*b**(19/2)*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 +
315*a**4*b**12*x**14) - 110*A*a**8*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x*
*10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 114*A*a**7*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(315*a**7*
b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 40*A*a**6*b**(25/2)*x**6*sqr
t(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) -
30*A*a**6*b**(11/2)*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 5*A
*a**5*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 +
315*a**4*b**12*x**14) - 66*A*a**5*b**(13/2)*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8
+ 105*a**3*b**6*x**10) + 30*A*a**4*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*
x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 34*A*a**4*b**(15/2)*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5
*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 40*A*a**3*b**(31/2)*x**12*sqrt(a/(b*x**2) + 1)/(315*a
**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 6*A*a**3*b**(17/2)*x**6*
sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 16*A*a**2*b**(33/2)*x**
14*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**
14) - 24*A*a**2*b**(19/2)*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x
**10) - 16*A*a*b**(21/2)*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x
**10) - A*b**(5/2)*sqrt(a/(b*x**2) + 1)/(5*x**4) - A*b**(7/2)*sqrt(a/(b*x**2) + 1)/(15*a*x**2) + 2*A*b**(9/2)*
sqrt(a/(b*x**2) + 1)/(15*a**2) - 15*B*a**7*b**(9/2)*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x
**8 + 105*a**3*b**6*x**10) - 33*B*a**6*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5
*x**8 + 105*a**3*b**6*x**10) - 17*B*a**5*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b*
*5*x**8 + 105*a**3*b**6*x**10) - 3*B*a**4*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b
**5*x**8 + 105*a**3*b**6*x**10) - 12*B*a**3*b**(17/2)*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4
*b**5*x**8 + 105*a**3*b**6*x**10) - 8*B*a**2*b**(19/2)*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a*
*4*b**5*x**8 + 105*a**3*b**6*x**10) - 2*B*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(5*x**4) - 7*B*b**(5/2)*sqrt(a/(b*x*
*2) + 1)/(15*x**2) - B*b**(7/2)*sqrt(a/(b*x**2) + 1)/(15*a)

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Giac [B]  time = 1.1743, size = 616, normalized size = 11.62 \begin{align*} \frac{2 \,{\left (63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} B b^{\frac{7}{2}} - 126 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} B a b^{\frac{7}{2}} + 126 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} A b^{\frac{9}{2}} + 378 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B a^{2} b^{\frac{7}{2}} + 210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} A a b^{\frac{9}{2}} - 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a^{3} b^{\frac{7}{2}} + 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A a^{2} b^{\frac{9}{2}} + 504 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{4} b^{\frac{7}{2}} + 378 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a^{3} b^{\frac{9}{2}} - 378 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{5} b^{\frac{7}{2}} + 378 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{4} b^{\frac{9}{2}} + 198 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{6} b^{\frac{7}{2}} + 54 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{5} b^{\frac{9}{2}} - 18 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{7} b^{\frac{7}{2}} + 18 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{6} b^{\frac{9}{2}} + 9 \, B a^{8} b^{\frac{7}{2}} - 2 \, A a^{7} b^{\frac{9}{2}}\right )}}{63 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)*(B*x^2+A)/x^10,x, algorithm="giac")

[Out]

2/63*(63*(sqrt(b)*x - sqrt(b*x^2 + a))^16*B*b^(7/2) - 126*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a*b^(7/2) + 126*(
sqrt(b)*x - sqrt(b*x^2 + a))^14*A*b^(9/2) + 378*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a^2*b^(7/2) + 210*(sqrt(b)*
x - sqrt(b*x^2 + a))^12*A*a*b^(9/2) - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^3*b^(7/2) + 630*(sqrt(b)*x - sq
rt(b*x^2 + a))^10*A*a^2*b^(9/2) + 504*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^4*b^(7/2) + 378*(sqrt(b)*x - sqrt(b*
x^2 + a))^8*A*a^3*b^(9/2) - 378*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^5*b^(7/2) + 378*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*A*a^4*b^(9/2) + 198*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^6*b^(7/2) + 54*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A
*a^5*b^(9/2) - 18*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^7*b^(7/2) + 18*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^6*b^(
9/2) + 9*B*a^8*b^(7/2) - 2*A*a^7*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9