3.585 \(\int \frac{x^6 (A+B x^2)}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{x^5 (4 A b-7 a B)}{12 b^2 \left (a+b x^2\right )^{3/2}}-\frac{5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt{a+b x^2}}+\frac{5 x \sqrt{a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{9/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}} \]

[Out]

-((4*A*b - 7*a*B)*x^5)/(12*b^2*(a + b*x^2)^(3/2)) + (B*x^7)/(4*b*(a + b*x^2)^(3/2)) - (5*(4*A*b - 7*a*B)*x^3)/
(12*b^3*Sqrt[a + b*x^2]) + (5*(4*A*b - 7*a*B)*x*Sqrt[a + b*x^2])/(8*b^4) - (5*a*(4*A*b - 7*a*B)*ArcTanh[(Sqrt[
b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0620309, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {459, 288, 321, 217, 206} \[ -\frac{x^5 (4 A b-7 a B)}{12 b^2 \left (a+b x^2\right )^{3/2}}-\frac{5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt{a+b x^2}}+\frac{5 x \sqrt{a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{9/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

-((4*A*b - 7*a*B)*x^5)/(12*b^2*(a + b*x^2)^(3/2)) + (B*x^7)/(4*b*(a + b*x^2)^(3/2)) - (5*(4*A*b - 7*a*B)*x^3)/
(12*b^3*Sqrt[a + b*x^2]) + (5*(4*A*b - 7*a*B)*x*Sqrt[a + b*x^2])/(8*b^4) - (5*a*(4*A*b - 7*a*B)*ArcTanh[(Sqrt[
b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2))

Rule 459

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

Rule 288

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

Rule 321

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac{(-4 A b+7 a B) \int \frac{x^6}{\left (a+b x^2\right )^{5/2}} \, dx}{4 b}\\ &=-\frac{(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}}+\frac{(5 (4 A b-7 a B)) \int \frac{x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{12 b^2}\\ &=-\frac{(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac{5 (4 A b-7 a B) x^3}{12 b^3 \sqrt{a+b x^2}}+\frac{(5 (4 A b-7 a B)) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{4 b^3}\\ &=-\frac{(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac{5 (4 A b-7 a B) x^3}{12 b^3 \sqrt{a+b x^2}}+\frac{5 (4 A b-7 a B) x \sqrt{a+b x^2}}{8 b^4}-\frac{(5 a (4 A b-7 a B)) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 b^4}\\ &=-\frac{(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac{5 (4 A b-7 a B) x^3}{12 b^3 \sqrt{a+b x^2}}+\frac{5 (4 A b-7 a B) x \sqrt{a+b x^2}}{8 b^4}-\frac{(5 a (4 A b-7 a B)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 b^4}\\ &=-\frac{(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac{B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac{5 (4 A b-7 a B) x^3}{12 b^3 \sqrt{a+b x^2}}+\frac{5 (4 A b-7 a B) x \sqrt{a+b x^2}}{8 b^4}-\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.230948, size = 139, normalized size = 0.93 \[ \frac{x \left (20 a^2 b \left (3 A-7 B x^2\right )-105 a^3 B+a b^2 x^2 \left (80 A-21 B x^2\right )+6 b^3 x^4 \left (2 A+B x^2\right )\right )}{24 b^4 \left (a+b x^2\right )^{3/2}}+\frac{5 \sqrt{a} \sqrt{a+b x^2} (7 a B-4 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2} \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(x*(-105*a^3*B + a*b^2*x^2*(80*A - 21*B*x^2) + 20*a^2*b*(3*A - 7*B*x^2) + 6*b^3*x^4*(2*A + B*x^2)))/(24*b^4*(a
+ b*x^2)^(3/2)) + (5*Sqrt[a]*(-4*A*b + 7*a*B)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(9/2)*Sqrt[1
+ (b*x^2)/a])

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 181, normalized size = 1.2 \begin{align*}{\frac{{x}^{7}B}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,Ba{x}^{5}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{a}^{2}B{x}^{3}}{24\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{a}^{2}Bx}{8\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{a}^{2}B}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{A{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,aA{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,aAx}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,Aa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x)

[Out]

1/4*B*x^7/b/(b*x^2+a)^(3/2)-7/8*B/b^2*a*x^5/(b*x^2+a)^(3/2)-35/24*B/b^3*a^2*x^3/(b*x^2+a)^(3/2)-35/8*B/b^4*a^2
*x/(b*x^2+a)^(1/2)+35/8*B/b^(9/2)*a^2*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/2*A*x^5/b/(b*x^2+a)^(3/2)+5/6*A/b^2*a*x^
3/(b*x^2+a)^(3/2)+5/2*A/b^3*a*x/(b*x^2+a)^(1/2)-5/2*A/b^(7/2)*a*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

________________________________________________________________________________________

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(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.75892, size = 856, normalized size = 5.74 \begin{align*} \left [-\frac{15 \,{\left (7 \, B a^{4} - 4 \, A a^{3} b +{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \,{\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (6 \, B b^{4} x^{7} - 3 \,{\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{5} - 20 \,{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \,{\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{48 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}, -\frac{15 \,{\left (7 \, B a^{4} - 4 \, A a^{3} b +{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \,{\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (6 \, B b^{4} x^{7} - 3 \,{\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{5} - 20 \,{\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \,{\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{24 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(15*(7*B*a^4 - 4*A*a^3*b + (7*B*a^2*b^2 - 4*A*a*b^3)*x^4 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x^2)*sqrt(b)*log
(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(6*B*b^4*x^7 - 3*(7*B*a*b^3 - 4*A*b^4)*x^5 - 20*(7*B*a^2*b^2
- 4*A*a*b^3)*x^3 - 15*(7*B*a^3*b - 4*A*a^2*b^2)*x)*sqrt(b*x^2 + a))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5), -1/24*(
15*(7*B*a^4 - 4*A*a^3*b + (7*B*a^2*b^2 - 4*A*a*b^3)*x^4 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x^2)*sqrt(-b)*arctan(sqr
t(-b)*x/sqrt(b*x^2 + a)) - (6*B*b^4*x^7 - 3*(7*B*a*b^3 - 4*A*b^4)*x^5 - 20*(7*B*a^2*b^2 - 4*A*a*b^3)*x^3 - 15*
(7*B*a^3*b - 4*A*a^2*b^2)*x)*sqrt(b*x^2 + a))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5)]

________________________________________________________________________________________

Sympy [B]  time = 29.4481, size = 804, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

A*(-15*a**(81/2)*b**22*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) +
6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) - 15*a**(79/2)*b**23*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/s
qrt(a))/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) + 15*a**40*
b**(45/2)*x/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) + 20*a*
*39*b**(47/2)*x**3/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a))
+ 3*a**38*b**(49/2)*x**5/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**
2/a))) + B*(105*a**(157/2)*b**41*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(24*a**(153/2)*b**(91/2)*sqrt(1 +
b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)) + 105*a**(155/2)*b**42*x**2*sqrt(1 + b*x**2/a)*a
sinh(sqrt(b)*x/sqrt(a))/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*
x**2/a)) - 105*a**78*b**(83/2)*x/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sq
rt(1 + b*x**2/a)) - 140*a**77*b**(85/2)*x**3/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(9
3/2)*x**2*sqrt(1 + b*x**2/a)) - 21*a**76*b**(87/2)*x**5/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(1
51/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)) + 6*a**75*b**(89/2)*x**7/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a)
+ 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)))

________________________________________________________________________________________

Giac [A]  time = 1.15069, size = 200, normalized size = 1.34 \begin{align*} \frac{{\left ({\left (3 \,{\left (\frac{2 \, B x^{2}}{b} - \frac{7 \, B a^{2} b^{5} - 4 \, A a b^{6}}{a b^{7}}\right )} x^{2} - \frac{20 \,{\left (7 \, B a^{3} b^{4} - 4 \, A a^{2} b^{5}\right )}}{a b^{7}}\right )} x^{2} - \frac{15 \,{\left (7 \, B a^{4} b^{3} - 4 \, A a^{3} b^{4}\right )}}{a b^{7}}\right )} x}{24 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{5 \,{\left (7 \, B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/24*((3*(2*B*x^2/b - (7*B*a^2*b^5 - 4*A*a*b^6)/(a*b^7))*x^2 - 20*(7*B*a^3*b^4 - 4*A*a^2*b^5)/(a*b^7))*x^2 - 1
5*(7*B*a^4*b^3 - 4*A*a^3*b^4)/(a*b^7))*x/(b*x^2 + a)^(3/2) - 5/8*(7*B*a^2 - 4*A*a*b)*log(abs(-sqrt(b)*x + sqrt
(b*x^2 + a)))/b^(9/2)