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

Optimal. Leaf size=320 \[ \frac{2 b^4 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{13 (a+b x)}+\frac{10 a b^3 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{11 (a+b x)}+\frac{20 a^2 b^2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{10 a^3 b x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)}+\frac{2 a^4 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{5 (a+b x)}+\frac{2 a^5 A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 b^5 B x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)} \]

[Out]

(2*a^5*A*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a^4*(5*A*b + a*B)*x^(5/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(5*(a + b*x)) + (10*a^3*b*(2*A*b + a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (
20*a^2*b^2*(A*b + a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)*x^(11/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(13*(a + b*x)) + (2*b^5*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 = 0.119564, antiderivative size = 320, 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} \[ \frac{2 b^4 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{13 (a+b x)}+\frac{10 a b^3 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{11 (a+b x)}+\frac{20 a^2 b^2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac{10 a^3 b x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)}+\frac{2 a^4 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{5 (a+b x)}+\frac{2 a^5 A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 b^5 B x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^5*A*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a^4*(5*A*b + a*B)*x^(5/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(5*(a + b*x)) + (10*a^3*b*(2*A*b + a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (
20*a^2*b^2*(A*b + a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)*x^(11/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(13*(a + b*x)) + (2*b^5*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 \sqrt{x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \sqrt{x} \left (a b+b^2 x\right )^5 (A+B x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a^5 A b^5 \sqrt{x}+a^4 b^5 (5 A b+a B) x^{3/2}+5 a^3 b^6 (2 A b+a B) x^{5/2}+10 a^2 b^7 (A b+a B) x^{7/2}+5 a b^8 (A b+2 a B) x^{9/2}+b^9 (A b+5 a B) x^{11/2}+b^{10} B x^{13/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{2 a^5 A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{2 a^4 (5 A b+a B) x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{10 a^3 b (2 A b+a B) x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{20 a^2 b^2 (A b+a B) x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{10 a b^3 (A b+2 a B) x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{2 b^4 (A b+5 a B) x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac{2 b^5 B x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0504346, size = 127, normalized size = 0.4 \[ \frac{2 x^{3/2} \sqrt{(a+b x)^2} \left (7150 a^3 b^2 x^2 (9 A+7 B x)+4550 a^2 b^3 x^3 (11 A+9 B x)+6435 a^4 b x (7 A+5 B x)+3003 a^5 (5 A+3 B x)+1575 a b^4 x^4 (13 A+11 B x)+231 b^5 x^5 (15 A+13 B x)\right )}{45045 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*Sqrt[(a + b*x)^2]*(3003*a^5*(5*A + 3*B*x) + 6435*a^4*b*x*(7*A + 5*B*x) + 7150*a^3*b^2*x^2*(9*A + 7*
B*x) + 4550*a^2*b^3*x^3*(11*A + 9*B*x) + 1575*a*b^4*x^4*(13*A + 11*B*x) + 231*b^5*x^5*(15*A + 13*B*x)))/(45045
*(a + b*x))

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Maple [A]  time = 0.007, size = 140, normalized size = 0.4 \begin{align*}{\frac{6006\,B{b}^{5}{x}^{6}+6930\,A{x}^{5}{b}^{5}+34650\,B{x}^{5}a{b}^{4}+40950\,A{x}^{4}a{b}^{4}+81900\,B{x}^{4}{a}^{2}{b}^{3}+100100\,A{x}^{3}{a}^{2}{b}^{3}+100100\,B{x}^{3}{a}^{3}{b}^{2}+128700\,A{x}^{2}{a}^{3}{b}^{2}+64350\,B{x}^{2}{a}^{4}b+90090\,A{a}^{4}bx+18018\,B{a}^{5}x+30030\,A{a}^{5}}{45045\, \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*x^(1/2),x)

[Out]

2/45045*x^(3/2)*(3003*B*b^5*x^6+3465*A*b^5*x^5+17325*B*a*b^4*x^5+20475*A*a*b^4*x^4+40950*B*a^2*b^3*x^4+50050*A
*a^2*b^3*x^3+50050*B*a^3*b^2*x^3+64350*A*a^3*b^2*x^2+32175*B*a^4*b*x^2+45045*A*a^4*b*x+9009*B*a^5*x+15015*A*a^
5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [A]  time = 1.1238, size = 325, normalized size = 1.02 \begin{align*} \frac{2}{45045} \,{\left (315 \,{\left (11 \, b^{5} x^{2} + 13 \, a b^{4} x\right )} x^{\frac{9}{2}} + 1820 \,{\left (9 \, a b^{4} x^{2} + 11 \, a^{2} b^{3} x\right )} x^{\frac{7}{2}} + 4290 \,{\left (7 \, a^{2} b^{3} x^{2} + 9 \, a^{3} b^{2} x\right )} x^{\frac{5}{2}} + 5148 \,{\left (5 \, a^{3} b^{2} x^{2} + 7 \, a^{4} b x\right )} x^{\frac{3}{2}} + 3003 \,{\left (3 \, a^{4} b x^{2} + 5 \, a^{5} x\right )} \sqrt{x}\right )} A + \frac{2}{45045} \,{\left (231 \,{\left (13 \, b^{5} x^{2} + 15 \, a b^{4} x\right )} x^{\frac{11}{2}} + 1260 \,{\left (11 \, a b^{4} x^{2} + 13 \, a^{2} b^{3} x\right )} x^{\frac{9}{2}} + 2730 \,{\left (9 \, a^{2} b^{3} x^{2} + 11 \, a^{3} b^{2} x\right )} x^{\frac{7}{2}} + 2860 \,{\left (7 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x\right )} x^{\frac{5}{2}} + 1287 \,{\left (5 \, a^{4} b x^{2} + 7 \, a^{5} x\right )} x^{\frac{3}{2}}\right )} B \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*x^(1/2),x, algorithm="maxima")

[Out]

2/45045*(315*(11*b^5*x^2 + 13*a*b^4*x)*x^(9/2) + 1820*(9*a*b^4*x^2 + 11*a^2*b^3*x)*x^(7/2) + 4290*(7*a^2*b^3*x
^2 + 9*a^3*b^2*x)*x^(5/2) + 5148*(5*a^3*b^2*x^2 + 7*a^4*b*x)*x^(3/2) + 3003*(3*a^4*b*x^2 + 5*a^5*x)*sqrt(x))*A
+ 2/45045*(231*(13*b^5*x^2 + 15*a*b^4*x)*x^(11/2) + 1260*(11*a*b^4*x^2 + 13*a^2*b^3*x)*x^(9/2) + 2730*(9*a^2*
b^3*x^2 + 11*a^3*b^2*x)*x^(7/2) + 2860*(7*a^3*b^2*x^2 + 9*a^4*b*x)*x^(5/2) + 1287*(5*a^4*b*x^2 + 7*a^5*x)*x^(3
/2))*B

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Fricas [A]  time = 1.45696, size = 297, normalized size = 0.93 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B b^{5} x^{7} + 15015 \, A a^{5} x + 3465 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + 20475 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + 50050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + 32175 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + 9009 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*x^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^5*x^7 + 15015*A*a^5*x + 3465*(5*B*a*b^4 + A*b^5)*x^6 + 20475*(2*B*a^2*b^3 + A*a*b^4)*x^5 + 5
0050*(B*a^3*b^2 + A*a^2*b^3)*x^4 + 32175*(B*a^4*b + 2*A*a^3*b^2)*x^3 + 9009*(B*a^5 + 5*A*a^4*b)*x^2)*sqrt(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)*x**(1/2),x)

[Out]

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

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Giac [A]  time = 1.14268, size = 266, normalized size = 0.83 \begin{align*} \frac{2}{15} \, B b^{5} x^{\frac{15}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{13} \, B a b^{4} x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{13} \, A b^{5} x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{11} \, B a^{2} b^{3} x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{11} \, A a b^{4} x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{9} \, B a^{3} b^{2} x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{9} \, A a^{2} b^{3} x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, B a^{4} b x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{7} \, A a^{3} b^{2} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{5} \, B a^{5} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + 2 \, A a^{4} b x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, A a^{5} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*x^(1/2),x, algorithm="giac")

[Out]

2/15*B*b^5*x^(15/2)*sgn(b*x + a) + 10/13*B*a*b^4*x^(13/2)*sgn(b*x + a) + 2/13*A*b^5*x^(13/2)*sgn(b*x + a) + 20
/11*B*a^2*b^3*x^(11/2)*sgn(b*x + a) + 10/11*A*a*b^4*x^(11/2)*sgn(b*x + a) + 20/9*B*a^3*b^2*x^(9/2)*sgn(b*x + a
) + 20/9*A*a^2*b^3*x^(9/2)*sgn(b*x + a) + 10/7*B*a^4*b*x^(7/2)*sgn(b*x + a) + 20/7*A*a^3*b^2*x^(7/2)*sgn(b*x +
a) + 2/5*B*a^5*x^(5/2)*sgn(b*x + a) + 2*A*a^4*b*x^(5/2)*sgn(b*x + a) + 2/3*A*a^5*x^(3/2)*sgn(b*x + a)