3.750 \(\int \sqrt{x} (A+B x) (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=159 \[ \frac{10}{11} a^2 b^3 x^{11/2} (4 a B+3 A b)+\frac{10}{9} a^3 b^2 x^{9/2} (3 a B+4 A b)+\frac{6}{7} a^4 b x^{7/2} (2 a B+5 A b)+\frac{2}{5} a^5 x^{5/2} (a B+6 A b)+\frac{2}{3} a^6 A x^{3/2}+\frac{2}{15} b^5 x^{15/2} (6 a B+A b)+\frac{6}{13} a b^4 x^{13/2} (5 a B+2 A b)+\frac{2}{17} b^6 B x^{17/2} \]

[Out]

(2*a^6*A*x^(3/2))/3 + (2*a^5*(6*A*b + a*B)*x^(5/2))/5 + (6*a^4*b*(5*A*b + 2*a*B)*x^(7/2))/7 + (10*a^3*b^2*(4*A
*b + 3*a*B)*x^(9/2))/9 + (10*a^2*b^3*(3*A*b + 4*a*B)*x^(11/2))/11 + (6*a*b^4*(2*A*b + 5*a*B)*x^(13/2))/13 + (2
*b^5*(A*b + 6*a*B)*x^(15/2))/15 + (2*b^6*B*x^(17/2))/17

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Rubi [A]  time = 0.0812867, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 76} \[ \frac{10}{11} a^2 b^3 x^{11/2} (4 a B+3 A b)+\frac{10}{9} a^3 b^2 x^{9/2} (3 a B+4 A b)+\frac{6}{7} a^4 b x^{7/2} (2 a B+5 A b)+\frac{2}{5} a^5 x^{5/2} (a B+6 A b)+\frac{2}{3} a^6 A x^{3/2}+\frac{2}{15} b^5 x^{15/2} (6 a B+A b)+\frac{6}{13} a b^4 x^{13/2} (5 a B+2 A b)+\frac{2}{17} b^6 B x^{17/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^6*A*x^(3/2))/3 + (2*a^5*(6*A*b + a*B)*x^(5/2))/5 + (6*a^4*b*(5*A*b + 2*a*B)*x^(7/2))/7 + (10*a^3*b^2*(4*A
*b + 3*a*B)*x^(9/2))/9 + (10*a^2*b^3*(3*A*b + 4*a*B)*x^(11/2))/11 + (6*a*b^4*(2*A*b + 5*a*B)*x^(13/2))/13 + (2
*b^5*(A*b + 6*a*B)*x^(15/2))/15 + (2*b^6*B*x^(17/2))/17

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 )^3 \, dx &=\int \sqrt{x} (a+b x)^6 (A+B x) \, dx\\ &=\int \left (a^6 A \sqrt{x}+a^5 (6 A b+a B) x^{3/2}+3 a^4 b (5 A b+2 a B) x^{5/2}+5 a^3 b^2 (4 A b+3 a B) x^{7/2}+5 a^2 b^3 (3 A b+4 a B) x^{9/2}+3 a b^4 (2 A b+5 a B) x^{11/2}+b^5 (A b+6 a B) x^{13/2}+b^6 B x^{15/2}\right ) \, dx\\ &=\frac{2}{3} a^6 A x^{3/2}+\frac{2}{5} a^5 (6 A b+a B) x^{5/2}+\frac{6}{7} a^4 b (5 A b+2 a B) x^{7/2}+\frac{10}{9} a^3 b^2 (4 A b+3 a B) x^{9/2}+\frac{10}{11} a^2 b^3 (3 A b+4 a B) x^{11/2}+\frac{6}{13} a b^4 (2 A b+5 a B) x^{13/2}+\frac{2}{15} b^5 (A b+6 a B) x^{15/2}+\frac{2}{17} b^6 B x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0827684, size = 103, normalized size = 0.65 \[ \frac{2 \left (\frac{x^{3/2} \left (96525 a^4 b^2 x^2+100100 a^3 b^3 x^3+61425 a^2 b^4 x^4+54054 a^5 b x+15015 a^6+20790 a b^5 x^5+3003 b^6 x^6\right ) (17 A b-3 a B)}{45045}+B x^{3/2} (a+b x)^7\right )}{17 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*x^(3/2)*(a + b*x)^7 + ((17*A*b - 3*a*B)*x^(3/2)*(15015*a^6 + 54054*a^5*b*x + 96525*a^4*b^2*x^2 + 100100*
a^3*b^3*x^3 + 61425*a^2*b^4*x^4 + 20790*a*b^5*x^5 + 3003*b^6*x^6))/45045))/(17*b)

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Maple [A]  time = 0.007, size = 148, normalized size = 0.9 \begin{align*}{\frac{90090\,B{b}^{6}{x}^{7}+102102\,A{b}^{6}{x}^{6}+612612\,B{x}^{6}a{b}^{5}+706860\,aA{b}^{5}{x}^{5}+1767150\,B{x}^{5}{a}^{2}{b}^{4}+2088450\,{a}^{2}A{b}^{4}{x}^{4}+2784600\,B{x}^{4}{a}^{3}{b}^{3}+3403400\,{a}^{3}A{b}^{3}{x}^{3}+2552550\,B{x}^{3}{a}^{4}{b}^{2}+3281850\,{a}^{4}A{b}^{2}{x}^{2}+1312740\,B{x}^{2}{a}^{5}b+1837836\,{a}^{5}Abx+306306\,B{a}^{6}x+510510\,A{a}^{6}}{765765}{x}^{{\frac{3}{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)^3*x^(1/2),x)

[Out]

2/765765*x^(3/2)*(45045*B*b^6*x^7+51051*A*b^6*x^6+306306*B*a*b^5*x^6+353430*A*a*b^5*x^5+883575*B*a^2*b^4*x^5+1
044225*A*a^2*b^4*x^4+1392300*B*a^3*b^3*x^4+1701700*A*a^3*b^3*x^3+1276275*B*a^4*b^2*x^3+1640925*A*a^4*b^2*x^2+6
56370*B*a^5*b*x^2+918918*A*a^5*b*x+153153*B*a^6*x+255255*A*a^6)

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Maxima [A]  time = 0.989634, size = 198, normalized size = 1.25 \begin{align*} \frac{2}{17} \, B b^{6} x^{\frac{17}{2}} + \frac{2}{3} \, A a^{6} x^{\frac{3}{2}} + \frac{2}{15} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{13}{2}} + \frac{10}{11} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{11}{2}} + \frac{10}{9} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac{5}{2}} \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)^3*x^(1/2),x, algorithm="maxima")

[Out]

2/17*B*b^6*x^(17/2) + 2/3*A*a^6*x^(3/2) + 2/15*(6*B*a*b^5 + A*b^6)*x^(15/2) + 6/13*(5*B*a^2*b^4 + 2*A*a*b^5)*x
^(13/2) + 10/11*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^(11/2) + 10/9*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^(9/2) + 6/7*(2*B*a^5
*b + 5*A*a^4*b^2)*x^(7/2) + 2/5*(B*a^6 + 6*A*a^5*b)*x^(5/2)

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Fricas [A]  time = 1.62096, size = 375, normalized size = 2.36 \begin{align*} \frac{2}{765765} \,{\left (45045 \, B b^{6} x^{8} + 255255 \, A a^{6} x + 51051 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + 176715 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} + 348075 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + 425425 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} + 328185 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + 153153 \,{\left (B a^{6} + 6 \, A a^{5} 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)^3*x^(1/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^6*x^8 + 255255*A*a^6*x + 51051*(6*B*a*b^5 + A*b^6)*x^7 + 176715*(5*B*a^2*b^4 + 2*A*a*b^5)*
x^6 + 348075*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^5 + 425425*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^4 + 328185*(2*B*a^5*b + 5*
A*a^4*b^2)*x^3 + 153153*(B*a^6 + 6*A*a^5*b)*x^2)*sqrt(x)

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Sympy [A]  time = 5.10497, size = 182, normalized size = 1.14 \begin{align*} \frac{2 A a^{6} x^{\frac{3}{2}}}{3} + \frac{2 B b^{6} x^{\frac{17}{2}}}{17} + \frac{2 x^{\frac{15}{2}} \left (A b^{6} + 6 B a b^{5}\right )}{15} + \frac{2 x^{\frac{13}{2}} \left (6 A a b^{5} + 15 B a^{2} b^{4}\right )}{13} + \frac{2 x^{\frac{11}{2}} \left (15 A a^{2} b^{4} + 20 B a^{3} b^{3}\right )}{11} + \frac{2 x^{\frac{9}{2}} \left (20 A a^{3} b^{3} + 15 B a^{4} b^{2}\right )}{9} + \frac{2 x^{\frac{7}{2}} \left (15 A a^{4} b^{2} + 6 B a^{5} b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (6 A a^{5} b + B a^{6}\right )}{5} \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)**3*x**(1/2),x)

[Out]

2*A*a**6*x**(3/2)/3 + 2*B*b**6*x**(17/2)/17 + 2*x**(15/2)*(A*b**6 + 6*B*a*b**5)/15 + 2*x**(13/2)*(6*A*a*b**5 +
15*B*a**2*b**4)/13 + 2*x**(11/2)*(15*A*a**2*b**4 + 20*B*a**3*b**3)/11 + 2*x**(9/2)*(20*A*a**3*b**3 + 15*B*a**
4*b**2)/9 + 2*x**(7/2)*(15*A*a**4*b**2 + 6*B*a**5*b)/7 + 2*x**(5/2)*(6*A*a**5*b + B*a**6)/5

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Giac [A]  time = 1.14641, size = 201, normalized size = 1.26 \begin{align*} \frac{2}{17} \, B b^{6} x^{\frac{17}{2}} + \frac{4}{5} \, B a b^{5} x^{\frac{15}{2}} + \frac{2}{15} \, A b^{6} x^{\frac{15}{2}} + \frac{30}{13} \, B a^{2} b^{4} x^{\frac{13}{2}} + \frac{12}{13} \, A a b^{5} x^{\frac{13}{2}} + \frac{40}{11} \, B a^{3} b^{3} x^{\frac{11}{2}} + \frac{30}{11} \, A a^{2} b^{4} x^{\frac{11}{2}} + \frac{10}{3} \, B a^{4} b^{2} x^{\frac{9}{2}} + \frac{40}{9} \, A a^{3} b^{3} x^{\frac{9}{2}} + \frac{12}{7} \, B a^{5} b x^{\frac{7}{2}} + \frac{30}{7} \, A a^{4} b^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B a^{6} x^{\frac{5}{2}} + \frac{12}{5} \, A a^{5} b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{6} x^{\frac{3}{2}} \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)^3*x^(1/2),x, algorithm="giac")

[Out]

2/17*B*b^6*x^(17/2) + 4/5*B*a*b^5*x^(15/2) + 2/15*A*b^6*x^(15/2) + 30/13*B*a^2*b^4*x^(13/2) + 12/13*A*a*b^5*x^
(13/2) + 40/11*B*a^3*b^3*x^(11/2) + 30/11*A*a^2*b^4*x^(11/2) + 10/3*B*a^4*b^2*x^(9/2) + 40/9*A*a^3*b^3*x^(9/2)
+ 12/7*B*a^5*b*x^(7/2) + 30/7*A*a^4*b^2*x^(7/2) + 2/5*B*a^6*x^(5/2) + 12/5*A*a^5*b*x^(5/2) + 2/3*A*a^6*x^(3/2
)