3.792 \(\int x^{5/2} (A+B x) (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=220 \[ \frac{2 b^2 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{13 (a+b x)}+\frac{6 a b x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac{2 a^2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{9 (a+b x)}+\frac{2 a^3 A x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{2 b^3 B x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)} \]

[Out]

(2*a^3*A*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*a^2*(3*A*b + a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(9*(a + b*x)) + (6*a*b*(A*b + a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (2*b
^2*(A*b + 3*a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^3*B*x^(15/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(15*(a + b*x))

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Rubi [A]  time = 0.0810922, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 76} \[ \frac{2 b^2 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{13 (a+b x)}+\frac{6 a b x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac{2 a^2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{9 (a+b x)}+\frac{2 a^3 A x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{2 b^3 B x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*a^3*A*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*a^2*(3*A*b + a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(9*(a + b*x)) + (6*a*b*(A*b + a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (2*b
^2*(A*b + 3*a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^3*B*x^(15/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(15*(a + b*x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x^{5/2} \left (a b+b^2 x\right )^3 (A+B x) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a^3 A b^3 x^{5/2}+a^2 b^3 (3 A b+a B) x^{7/2}+3 a b^4 (A b+a B) x^{9/2}+b^5 (A b+3 a B) x^{11/2}+b^6 B x^{13/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{2 a^3 A x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{2 a^2 (3 A b+a B) x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{6 a b (A b+a B) x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{2 b^2 (A b+3 a B) x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac{2 b^3 B x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0359592, size = 89, normalized size = 0.4 \[ \frac{2 x^{7/2} \sqrt{(a+b x)^2} \left (1365 a^2 b x (11 A+9 B x)+715 a^3 (9 A+7 B x)+945 a b^2 x^2 (13 A+11 B x)+231 b^3 x^3 (15 A+13 B x)\right )}{45045 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*x^(7/2)*Sqrt[(a + b*x)^2]*(715*a^3*(9*A + 7*B*x) + 1365*a^2*b*x*(11*A + 9*B*x) + 945*a*b^2*x^2*(13*A + 11*B
*x) + 231*b^3*x^3*(15*A + 13*B*x)))/(45045*(a + b*x))

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Maple [A]  time = 0.007, size = 92, normalized size = 0.4 \begin{align*}{\frac{6006\,B{x}^{4}{b}^{3}+6930\,A{b}^{3}{x}^{3}+20790\,B{x}^{3}a{b}^{2}+24570\,A{x}^{2}a{b}^{2}+24570\,B{x}^{2}{a}^{2}b+30030\,A{a}^{2}bx+10010\,{a}^{3}Bx+12870\,A{a}^{3}}{45045\, \left ( bx+a \right ) ^{3}}{x}^{{\frac{7}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

2/45045*x^(7/2)*(3003*B*b^3*x^4+3465*A*b^3*x^3+10395*B*a*b^2*x^3+12285*A*a*b^2*x^2+12285*B*a^2*b*x^2+15015*A*a
^2*b*x+5005*B*a^3*x+6435*A*a^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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Maxima [A]  time = 1.10637, size = 185, normalized size = 0.84 \begin{align*} \frac{2}{9009} \,{\left (63 \,{\left (11 \, b^{3} x^{2} + 13 \, a b^{2} x\right )} x^{\frac{9}{2}} + 182 \,{\left (9 \, a b^{2} x^{2} + 11 \, a^{2} b x\right )} x^{\frac{7}{2}} + 143 \,{\left (7 \, a^{2} b x^{2} + 9 \, a^{3} x\right )} x^{\frac{5}{2}}\right )} A + \frac{2}{6435} \,{\left (33 \,{\left (13 \, b^{3} x^{2} + 15 \, a b^{2} x\right )} x^{\frac{11}{2}} + 90 \,{\left (11 \, a b^{2} x^{2} + 13 \, a^{2} b x\right )} x^{\frac{9}{2}} + 65 \,{\left (9 \, a^{2} b x^{2} + 11 \, a^{3} x\right )} x^{\frac{7}{2}}\right )} B \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

2/9009*(63*(11*b^3*x^2 + 13*a*b^2*x)*x^(9/2) + 182*(9*a*b^2*x^2 + 11*a^2*b*x)*x^(7/2) + 143*(7*a^2*b*x^2 + 9*a
^3*x)*x^(5/2))*A + 2/6435*(33*(13*b^3*x^2 + 15*a*b^2*x)*x^(11/2) + 90*(11*a*b^2*x^2 + 13*a^2*b*x)*x^(9/2) + 65
*(9*a^2*b*x^2 + 11*a^3*x)*x^(7/2))*B

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Fricas [A]  time = 1.58552, size = 196, normalized size = 0.89 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B b^{3} x^{7} + 6435 \, A a^{3} x^{3} + 3465 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 12285 \,{\left (B a^{2} b + A a b^{2}\right )} x^{5} + 5005 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^3*x^7 + 6435*A*a^3*x^3 + 3465*(3*B*a*b^2 + A*b^3)*x^6 + 12285*(B*a^2*b + A*a*b^2)*x^5 + 5005
*(B*a^3 + 3*A*a^2*b)*x^4)*sqrt(x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16331, size = 169, normalized size = 0.77 \begin{align*} \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{13} \, B a b^{2} x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{13} \, A b^{3} x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{11} \, B a^{2} b x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{11} \, A a b^{2} x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

2/15*B*b^3*x^(15/2)*sgn(b*x + a) + 6/13*B*a*b^2*x^(13/2)*sgn(b*x + a) + 2/13*A*b^3*x^(13/2)*sgn(b*x + a) + 6/1
1*B*a^2*b*x^(11/2)*sgn(b*x + a) + 6/11*A*a*b^2*x^(11/2)*sgn(b*x + a) + 2/9*B*a^3*x^(9/2)*sgn(b*x + a) + 2/3*A*
a^2*b*x^(9/2)*sgn(b*x + a) + 2/7*A*a^3*x^(7/2)*sgn(b*x + a)