\(\int x (A+B x) (a+b x^2)^{5/2} \, dx\) [17]

Optimal result

Integrand size = 18, antiderivative size = 126 \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=-\frac {5 a^3 B x \sqrt {a+b x^2}}{128 b}-\frac {5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}-\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {(8 A+7 B x) \left (a+b x^2\right )^{7/2}}{56 b}-\frac {5 a^4 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \]

[Out]

-5/192*a^2*B*x*(b*x^2+a)^(3/2)/b-1/48*a*B*x*(b*x^2+a)^(5/2)/b+1/56*(7*B*x+8*A)*(b*x^2+a)^(7/2)/b-5/128*a^4*B*a
rctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(3/2)-5/128*a^3*B*x*(b*x^2+a)^(1/2)/b

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {794, 201, 223, 212} \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=-\frac {5 a^4 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}}-\frac {5 a^3 B x \sqrt {a+b x^2}}{128 b}-\frac {5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {\left (a+b x^2\right )^{7/2} (8 A+7 B x)}{56 b}-\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b} \]

[In]

Int[x*(A + B*x)*(a + b*x^2)^(5/2),x]

[Out]

(-5*a^3*B*x*Sqrt[a + b*x^2])/(128*b) - (5*a^2*B*x*(a + b*x^2)^(3/2))/(192*b) - (a*B*x*(a + b*x^2)^(5/2))/(48*b
) + ((8*A + 7*B*x)*(a + b*x^2)^(7/2))/(56*b) - (5*a^4*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(3/2))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

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

Rule 223

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(8 A+7 B x) \left (a+b x^2\right )^{7/2}}{56 b}-\frac {(a B) \int \left (a+b x^2\right )^{5/2} \, dx}{8 b} \\ & = -\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {(8 A+7 B x) \left (a+b x^2\right )^{7/2}}{56 b}-\frac {\left (5 a^2 B\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{48 b} \\ & = -\frac {5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}-\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {(8 A+7 B x) \left (a+b x^2\right )^{7/2}}{56 b}-\frac {\left (5 a^3 B\right ) \int \sqrt {a+b x^2} \, dx}{64 b} \\ & = -\frac {5 a^3 B x \sqrt {a+b x^2}}{128 b}-\frac {5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}-\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {(8 A+7 B x) \left (a+b x^2\right )^{7/2}}{56 b}-\frac {\left (5 a^4 B\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b} \\ & = -\frac {5 a^3 B x \sqrt {a+b x^2}}{128 b}-\frac {5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}-\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {(8 A+7 B x) \left (a+b x^2\right )^{7/2}}{56 b}-\frac {\left (5 a^4 B\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b} \\ & = -\frac {5 a^3 B x \sqrt {a+b x^2}}{128 b}-\frac {5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}-\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {(8 A+7 B x) \left (a+b x^2\right )^{7/2}}{56 b}-\frac {5 a^4 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94 \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (48 b^3 x^6 (8 A+7 B x)+3 a^3 (128 A+35 B x)+8 a b^2 x^4 (144 A+119 B x)+2 a^2 b x^2 (576 A+413 B x)\right )+105 a^4 B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2688 b^{3/2}} \]

[In]

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

[Out]

(Sqrt[b]*Sqrt[a + b*x^2]*(48*b^3*x^6*(8*A + 7*B*x) + 3*a^3*(128*A + 35*B*x) + 8*a*b^2*x^4*(144*A + 119*B*x) +
2*a^2*b*x^2*(576*A + 413*B*x)) + 105*a^4*B*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2688*b^(3/2))

Maple [A] (verified)

Time = 3.53 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86

method	result	size
default	\(B \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )+\frac {A \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}\)	\(108\)
risch	\(\frac {\left (336 b^{3} B \,x^{7}+384 x^{6} b^{3} A +952 B a \,b^{2} x^{5}+1152 a A \,b^{2} x^{4}+826 B \,a^{2} b \,x^{3}+1152 a^{2} A b \,x^{2}+105 a^{3} B x +384 a^{3} A \right ) \sqrt {b \,x^{2}+a}}{2688 b}-\frac {5 B \,a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}\)	\(113\)

[In]

int(x*(B*x+A)*(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.01 \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\left [\frac {105 \, B a^{4} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (336 \, B b^{4} x^{7} + 384 \, A b^{4} x^{6} + 952 \, B a b^{3} x^{5} + 1152 \, A a b^{3} x^{4} + 826 \, B a^{2} b^{2} x^{3} + 1152 \, A a^{2} b^{2} x^{2} + 105 \, B a^{3} b x + 384 \, A a^{3} b\right )} \sqrt {b x^{2} + a}}{5376 \, b^{2}}, \frac {105 \, B a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (336 \, B b^{4} x^{7} + 384 \, A b^{4} x^{6} + 952 \, B a b^{3} x^{5} + 1152 \, A a b^{3} x^{4} + 826 \, B a^{2} b^{2} x^{3} + 1152 \, A a^{2} b^{2} x^{2} + 105 \, B a^{3} b x + 384 \, A a^{3} b\right )} \sqrt {b x^{2} + a}}{2688 \, b^{2}}\right ] \]

[In]

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

[Out]

[1/5376*(105*B*a^4*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(336*B*b^4*x^7 + 384*A*b^4*x^6
+ 952*B*a*b^3*x^5 + 1152*A*a*b^3*x^4 + 826*B*a^2*b^2*x^3 + 1152*A*a^2*b^2*x^2 + 105*B*a^3*b*x + 384*A*a^3*b)*s
qrt(b*x^2 + a))/b^2, 1/2688*(105*B*a^4*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (336*B*b^4*x^7 + 384*A*b^
4*x^6 + 952*B*a*b^3*x^5 + 1152*A*a*b^3*x^4 + 826*B*a^2*b^2*x^3 + 1152*A*a^2*b^2*x^2 + 105*B*a^3*b*x + 384*A*a^
3*b)*sqrt(b*x^2 + a))/b^2]

Sympy [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.33 \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\begin {cases} - \frac {5 B a^{4} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{128 b} + \sqrt {a + b x^{2}} \left (\frac {A a^{3}}{7 b} + \frac {3 A a^{2} x^{2}}{7} + \frac {3 A a b x^{4}}{7} + \frac {A b^{2} x^{6}}{7} + \frac {5 B a^{3} x}{128 b} + \frac {59 B a^{2} x^{3}}{192} + \frac {17 B a b x^{5}}{48} + \frac {B b^{2} x^{7}}{8}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{2}}{2} + \frac {B x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-5*B*a**4*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x
**2), True))/(128*b) + sqrt(a + b*x**2)*(A*a**3/(7*b) + 3*A*a**2*x**2/7 + 3*A*a*b*x**4/7 + A*b**2*x**6/7 + 5*B
*a**3*x/(128*b) + 59*B*a**2*x**3/192 + 17*B*a*b*x**5/48 + B*b**2*x**7/8), Ne(b, 0)), (a**(5/2)*(A*x**2/2 + B*x
**3/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} B a^{3} x}{128 \, b} - \frac {5 \, B a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{7 \, b} \]

[In]

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

[Out]

1/8*(b*x^2 + a)^(7/2)*B*x/b - 1/48*(b*x^2 + a)^(5/2)*B*a*x/b - 5/192*(b*x^2 + a)^(3/2)*B*a^2*x/b - 5/128*sqrt(
b*x^2 + a)*B*a^3*x/b - 5/128*B*a^4*arcsinh(b*x/sqrt(a*b))/b^(3/2) + 1/7*(b*x^2 + a)^(7/2)*A/b

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\frac {5 \, B a^{4} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} + \frac {1}{2688} \, {\left (\frac {384 \, A a^{3}}{b} + {\left (\frac {105 \, B a^{3}}{b} + 2 \, {\left (576 \, A a^{2} + {\left (413 \, B a^{2} + 4 \, {\left (144 \, A a b + {\left (119 \, B a b + 6 \, {\left (7 \, B b^{2} x + 8 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {b x^{2} + a} \]

[In]

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

[Out]

5/128*B*a^4*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + 1/2688*(384*A*a^3/b + (105*B*a^3/b + 2*(576*A*a^2
+ (413*B*a^2 + 4*(144*A*a*b + (119*B*a*b + 6*(7*B*b^2*x + 8*A*b^2)*x)*x)*x)*x)*x)*x)*sqrt(b*x^2 + a)

Mupad [F(-1)]

Timed out. \[ \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\int x\,{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]

[In]

int(x*(a + b*x^2)^(5/2)*(A + B*x),x)

[Out]

int(x*(a + b*x^2)^(5/2)*(A + B*x), x)