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

Optimal. Leaf size=126 \[ -\frac {5 a^4 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}}-\frac {5 a^3 B x \sqrt {a+b x^2}}{128 b}-\frac {5 a^2 B x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {\left (a+b x^2\right )^{7/2} (8 A+7 B x)}{56 b}-\frac {a B x \left (a+b x^2\right )^{5/2}}{48 b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {780, 195, 217, 206} \begin {gather*} -\frac {5 a^4 B \tanh ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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 195

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 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])

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 780

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*} \int x (A+B x) \left (a+b x^2\right )^{5/2} \, dx &=\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 ) \operatorname {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]  time = 0.55, size = 112, normalized size = 0.89 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-\frac {7 a B x \left (\frac {15 a^{7/2} \sqrt {\frac {b x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} x}+\left (a+b x^2\right ) \left (33 a^2+26 a b x^2+8 b^2 x^4\right )\right )}{\left (a+b x^2\right )^4}+384 A+336 B x\right )}{2688 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^2)^(7/2)*(384*A + 336*B*x - (7*a*B*x*((a + b*x^2)*(33*a^2 + 26*a*b*x^2 + 8*b^2*x^4) + (15*a^(7/2)*Sq
rt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*x)))/(a + b*x^2)^4))/(2688*b)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.43, size = 125, normalized size = 0.99 \begin {gather*} \frac {5 a^4 B \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{128 b^{3/2}}+\frac {\sqrt {a+b x^2} \left (384 a^3 A+105 a^3 B x+1152 a^2 A b x^2+826 a^2 b B x^3+1152 a A b^2 x^4+952 a b^2 B x^5+384 A b^3 x^6+336 b^3 B x^7\right )}{2688 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x^2]*(384*a^3*A + 105*a^3*B*x + 1152*a^2*A*b*x^2 + 826*a^2*b*B*x^3 + 1152*a*A*b^2*x^4 + 952*a*b^2*
B*x^5 + 384*A*b^3*x^6 + 336*b^3*B*x^7))/(2688*b) + (5*a^4*B*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(128*b^(3/2))

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 253, normalized size = 2.01 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

________________________________________________________________________________________

giac [A]  time = 0.50, size = 114, normalized size = 0.90 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 113, normalized size = 0.90 \begin {gather*} -\frac {5 B \,a^{4} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}-\frac {5 \sqrt {b \,x^{2}+a}\, B \,a^{3} x}{128 b}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,a^{2} x}{192 b}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B a x}{48 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} B x}{8 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} A}{7 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.44, size = 105, normalized size = 0.83 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 26.54, size = 354, normalized size = 2.81 \begin {gather*} A a^{2} \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) + 2 A a b \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + b x^{2}}}{15 b^{2}} + \frac {a x^{2} \sqrt {a + b x^{2}}}{15 b} + \frac {x^{4} \sqrt {a + b x^{2}}}{5} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + A b^{2} \left (\begin {cases} \frac {8 a^{3} \sqrt {a + b x^{2}}}{105 b^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + b x^{2}}}{105 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{2}}}{35 b} + \frac {x^{6} \sqrt {a + b x^{2}}}{7} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \frac {5 B a^{\frac {7}{2}} x}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {133 B a^{\frac {5}{2}} x^{3}}{384 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {127 B a^{\frac {3}{2}} b x^{5}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 B \sqrt {a} b^{2} x^{7}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 B a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {3}{2}}} + \frac {B b^{3} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*Piecewise((sqrt(a)*x**2/2, Eq(b, 0)), ((a + b*x**2)**(3/2)/(3*b), True)) + 2*A*a*b*Piecewise((-2*a**2*s
qrt(a + b*x**2)/(15*b**2) + a*x**2*sqrt(a + b*x**2)/(15*b) + x**4*sqrt(a + b*x**2)/5, Ne(b, 0)), (sqrt(a)*x**4
/4, True)) + A*b**2*Piecewise((8*a**3*sqrt(a + b*x**2)/(105*b**3) - 4*a**2*x**2*sqrt(a + b*x**2)/(105*b**2) +
a*x**4*sqrt(a + b*x**2)/(35*b) + x**6*sqrt(a + b*x**2)/7, Ne(b, 0)), (sqrt(a)*x**6/6, True)) + 5*B*a**(7/2)*x/
(128*b*sqrt(1 + b*x**2/a)) + 133*B*a**(5/2)*x**3/(384*sqrt(1 + b*x**2/a)) + 127*B*a**(3/2)*b*x**5/(192*sqrt(1
+ b*x**2/a)) + 23*B*sqrt(a)*b**2*x**7/(48*sqrt(1 + b*x**2/a)) - 5*B*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**(3/2
)) + B*b**3*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

________________________________________________________________________________________