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

Optimal. Leaf size=157 \[ \frac{10}{9} a^2 b^3 x^{9/2} (4 a B+3 A b)+\frac{10}{7} a^3 b^2 x^{7/2} (3 a B+4 A b)+\frac{6}{5} a^4 b x^{5/2} (2 a B+5 A b)+\frac{2}{3} a^5 x^{3/2} (a B+6 A b)+2 a^6 A \sqrt{x}+\frac{2}{13} b^5 x^{13/2} (6 a B+A b)+\frac{6}{11} a b^4 x^{11/2} (5 a B+2 A b)+\frac{2}{15} b^6 B x^{15/2} \]

[Out]

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

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Rubi [A]  time = 0.0798598, antiderivative size = 157, 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} \[ \frac{10}{9} a^2 b^3 x^{9/2} (4 a B+3 A b)+\frac{10}{7} a^3 b^2 x^{7/2} (3 a B+4 A b)+\frac{6}{5} a^4 b x^{5/2} (2 a B+5 A b)+\frac{2}{3} a^5 x^{3/2} (a B+6 A b)+2 a^6 A \sqrt{x}+\frac{2}{13} b^5 x^{13/2} (6 a B+A b)+\frac{6}{11} a b^4 x^{11/2} (5 a B+2 A b)+\frac{2}{15} b^6 B x^{15/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0852304, size = 103, normalized size = 0.66 \[ \frac{2 \left (\frac{\sqrt{x} \left (9009 a^4 b^2 x^2+8580 a^3 b^3 x^3+5005 a^2 b^4 x^4+6006 a^5 b x+3003 a^6+1638 a b^5 x^5+231 b^6 x^6\right ) (15 A b-a B)}{3003}+B \sqrt{x} (a+b x)^7\right )}{15 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*Sqrt[x]*(a + b*x)^7 + ((15*A*b - a*B)*Sqrt[x]*(3003*a^6 + 6006*a^5*b*x + 9009*a^4*b^2*x^2 + 8580*a^3*b^3
*x^3 + 5005*a^2*b^4*x^4 + 1638*a*b^5*x^5 + 231*b^6*x^6))/3003))/(15*b)

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Maple [A]  time = 0.009, size = 148, normalized size = 0.9 \begin{align*}{\frac{6006\,B{b}^{6}{x}^{7}+6930\,A{b}^{6}{x}^{6}+41580\,B{x}^{6}a{b}^{5}+49140\,aA{b}^{5}{x}^{5}+122850\,B{x}^{5}{a}^{2}{b}^{4}+150150\,{a}^{2}A{b}^{4}{x}^{4}+200200\,B{x}^{4}{a}^{3}{b}^{3}+257400\,{a}^{3}A{b}^{3}{x}^{3}+193050\,B{x}^{3}{a}^{4}{b}^{2}+270270\,{a}^{4}A{b}^{2}{x}^{2}+108108\,B{x}^{2}{a}^{5}b+180180\,{a}^{5}Abx+30030\,B{a}^{6}x+90090\,A{a}^{6}}{45045}\sqrt{x}} \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]

2/45045*x^(1/2)*(3003*B*b^6*x^7+3465*A*b^6*x^6+20790*B*a*b^5*x^6+24570*A*a*b^5*x^5+61425*B*a^2*b^4*x^5+75075*A
*a^2*b^4*x^4+100100*B*a^3*b^3*x^4+128700*A*a^3*b^3*x^3+96525*B*a^4*b^2*x^3+135135*A*a^4*b^2*x^2+54054*B*a^5*b*
x^2+90090*A*a^5*b*x+15015*B*a^6*x+45045*A*a^6)

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Maxima [A]  time = 1.03086, size = 198, normalized size = 1.26 \begin{align*} \frac{2}{15} \, B b^{6} x^{\frac{15}{2}} + 2 \, A a^{6} \sqrt{x} + \frac{2}{13} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{11}{2}} + \frac{10}{9} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{9}{2}} + \frac{10}{7} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} 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="maxima")

[Out]

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

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Fricas [A]  time = 1.48393, size = 358, normalized size = 2.28 \begin{align*} \frac{2}{45045} \,{\left (3003 \, B b^{6} x^{7} + 45045 \, A a^{6} + 3465 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 12285 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 25025 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 32175 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 27027 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 15015 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\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]

2/45045*(3003*B*b^6*x^7 + 45045*A*a^6 + 3465*(6*B*a*b^5 + A*b^6)*x^6 + 12285*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 2
5025*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 32175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 27027*(2*B*a^5*b + 5*A*a^4*b^2)
*x^2 + 15015*(B*a^6 + 6*A*a^5*b)*x)*sqrt(x)

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Sympy [A]  time = 4.3166, size = 211, normalized size = 1.34 \begin{align*} 2 A a^{6} \sqrt{x} + 4 A a^{5} b x^{\frac{3}{2}} + 6 A a^{4} b^{2} x^{\frac{5}{2}} + \frac{40 A a^{3} b^{3} x^{\frac{7}{2}}}{7} + \frac{10 A a^{2} b^{4} x^{\frac{9}{2}}}{3} + \frac{12 A a b^{5} x^{\frac{11}{2}}}{11} + \frac{2 A b^{6} x^{\frac{13}{2}}}{13} + \frac{2 B a^{6} x^{\frac{3}{2}}}{3} + \frac{12 B a^{5} b x^{\frac{5}{2}}}{5} + \frac{30 B a^{4} b^{2} x^{\frac{7}{2}}}{7} + \frac{40 B a^{3} b^{3} x^{\frac{9}{2}}}{9} + \frac{30 B a^{2} b^{4} x^{\frac{11}{2}}}{11} + \frac{12 B a b^{5} x^{\frac{13}{2}}}{13} + \frac{2 B b^{6} x^{\frac{15}{2}}}{15} \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*sqrt(x) + 4*A*a**5*b*x**(3/2) + 6*A*a**4*b**2*x**(5/2) + 40*A*a**3*b**3*x**(7/2)/7 + 10*A*a**2*b**4*x
**(9/2)/3 + 12*A*a*b**5*x**(11/2)/11 + 2*A*b**6*x**(13/2)/13 + 2*B*a**6*x**(3/2)/3 + 12*B*a**5*b*x**(5/2)/5 +
30*B*a**4*b**2*x**(7/2)/7 + 40*B*a**3*b**3*x**(9/2)/9 + 30*B*a**2*b**4*x**(11/2)/11 + 12*B*a*b**5*x**(13/2)/13
+ 2*B*b**6*x**(15/2)/15

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Giac [A]  time = 1.17948, size = 201, normalized size = 1.28 \begin{align*} \frac{2}{15} \, B b^{6} x^{\frac{15}{2}} + \frac{12}{13} \, B a b^{5} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{6} x^{\frac{13}{2}} + \frac{30}{11} \, B a^{2} b^{4} x^{\frac{11}{2}} + \frac{12}{11} \, A a b^{5} x^{\frac{11}{2}} + \frac{40}{9} \, B a^{3} b^{3} x^{\frac{9}{2}} + \frac{10}{3} \, A a^{2} b^{4} x^{\frac{9}{2}} + \frac{30}{7} \, B a^{4} b^{2} x^{\frac{7}{2}} + \frac{40}{7} \, A a^{3} b^{3} x^{\frac{7}{2}} + \frac{12}{5} \, B a^{5} b x^{\frac{5}{2}} + 6 \, A a^{4} b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{6} x^{\frac{3}{2}} + 4 \, A a^{5} b x^{\frac{3}{2}} + 2 \, A a^{6} \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="giac")

[Out]

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