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

Optimal. Leaf size=151 \[ \frac{10}{7} a^2 b^3 x^{7/2} (4 a B+3 A b)+2 a^3 b^2 x^{5/2} (3 a B+4 A b)+2 a^4 b x^{3/2} (2 a B+5 A b)+2 a^5 \sqrt{x} (a B+6 A b)-\frac{2 a^6 A}{\sqrt{x}}+\frac{2}{11} b^5 x^{11/2} (6 a B+A b)+\frac{2}{3} a b^4 x^{9/2} (5 a B+2 A b)+\frac{2}{13} b^6 B x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0794442, antiderivative size = 151, 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} \[ \frac{10}{7} a^2 b^3 x^{7/2} (4 a B+3 A b)+2 a^3 b^2 x^{5/2} (3 a B+4 A b)+2 a^4 b x^{3/2} (2 a B+5 A b)+2 a^5 \sqrt{x} (a B+6 A b)-\frac{2 a^6 A}{\sqrt{x}}+\frac{2}{11} b^5 x^{11/2} (6 a B+A b)+\frac{2}{3} a b^4 x^{9/2} (5 a B+2 A b)+\frac{2}{13} b^6 B x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.073624, size = 101, normalized size = 0.67 \[ \frac{2 \left (\frac{\sqrt{x} \left (9009 a^4 b^2 x^2+8580 a^3 b^3 x^3+5005 a^2 b^4 x^4+6006 a^5 b x+3003 a^6+1638 a b^5 x^5+231 b^6 x^6\right ) (a B+13 A b)}{3003}-\frac{A (a+b x)^7}{\sqrt{x}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-((A*(a + b*x)^7)/Sqrt[x]) + ((13*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))/a

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Maple [A]  time = 0.007, size = 148, normalized size = 1. \begin{align*} -{\frac{-462\,B{b}^{6}{x}^{7}-546\,A{b}^{6}{x}^{6}-3276\,B{x}^{6}a{b}^{5}-4004\,aA{b}^{5}{x}^{5}-10010\,B{x}^{5}{a}^{2}{b}^{4}-12870\,{a}^{2}A{b}^{4}{x}^{4}-17160\,B{x}^{4}{a}^{3}{b}^{3}-24024\,{a}^{3}A{b}^{3}{x}^{3}-18018\,B{x}^{3}{a}^{4}{b}^{2}-30030\,{a}^{4}A{b}^{2}{x}^{2}-12012\,B{x}^{2}{a}^{5}b-36036\,{a}^{5}Abx-6006\,B{a}^{6}x+6006\,A{a}^{6}}{3003}{\frac{1}{\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^(3/2),x)

[Out]

-2/3003*(-231*B*b^6*x^7-273*A*b^6*x^6-1638*B*a*b^5*x^6-2002*A*a*b^5*x^5-5005*B*a^2*b^4*x^5-6435*A*a^2*b^4*x^4-
8580*B*a^3*b^3*x^4-12012*A*a^3*b^3*x^3-9009*B*a^4*b^2*x^3-15015*A*a^4*b^2*x^2-6006*B*a^5*b*x^2-18018*A*a^5*b*x
-3003*B*a^6*x+3003*A*a^6)/x^(1/2)

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Maxima [A]  time = 1.0105, size = 198, normalized size = 1.31 \begin{align*} \frac{2}{13} \, B b^{6} x^{\frac{13}{2}} - \frac{2 \, A a^{6}}{\sqrt{x}} + \frac{2}{11} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{11}{2}} + \frac{2}{3} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{9}{2}} + \frac{10}{7} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{7}{2}} + 2 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{5}{2}} + 2 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{6} + 6 \, A a^{5} b\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^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.51245, size = 346, normalized size = 2.29 \begin{align*} \frac{2 \,{\left (231 \, B b^{6} x^{7} - 3003 \, A a^{6} + 273 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 1001 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 2145 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 3003 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3003 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3003 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{3003 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/3003*(231*B*b^6*x^7 - 3003*A*a^6 + 273*(6*B*a*b^5 + A*b^6)*x^6 + 1001*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 2145*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 3003*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3003*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 3
003*(B*a^6 + 6*A*a^5*b)*x)/sqrt(x)

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Sympy [A]  time = 4.88776, size = 204, normalized size = 1.35 \begin{align*} - \frac{2 A a^{6}}{\sqrt{x}} + 12 A a^{5} b \sqrt{x} + 10 A a^{4} b^{2} x^{\frac{3}{2}} + 8 A a^{3} b^{3} x^{\frac{5}{2}} + \frac{30 A a^{2} b^{4} x^{\frac{7}{2}}}{7} + \frac{4 A a b^{5} x^{\frac{9}{2}}}{3} + \frac{2 A b^{6} x^{\frac{11}{2}}}{11} + 2 B a^{6} \sqrt{x} + 4 B a^{5} b x^{\frac{3}{2}} + 6 B a^{4} b^{2} x^{\frac{5}{2}} + \frac{40 B a^{3} b^{3} x^{\frac{7}{2}}}{7} + \frac{10 B a^{2} b^{4} x^{\frac{9}{2}}}{3} + \frac{12 B a b^{5} x^{\frac{11}{2}}}{11} + \frac{2 B b^{6} x^{\frac{13}{2}}}{13} \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**(3/2),x)

[Out]

-2*A*a**6/sqrt(x) + 12*A*a**5*b*sqrt(x) + 10*A*a**4*b**2*x**(3/2) + 8*A*a**3*b**3*x**(5/2) + 30*A*a**2*b**4*x*
*(7/2)/7 + 4*A*a*b**5*x**(9/2)/3 + 2*A*b**6*x**(11/2)/11 + 2*B*a**6*sqrt(x) + 4*B*a**5*b*x**(3/2) + 6*B*a**4*b
**2*x**(5/2) + 40*B*a**3*b**3*x**(7/2)/7 + 10*B*a**2*b**4*x**(9/2)/3 + 12*B*a*b**5*x**(11/2)/11 + 2*B*b**6*x**
(13/2)/13

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Giac [A]  time = 1.14489, size = 201, normalized size = 1.33 \begin{align*} \frac{2}{13} \, B b^{6} x^{\frac{13}{2}} + \frac{12}{11} \, B a b^{5} x^{\frac{11}{2}} + \frac{2}{11} \, A b^{6} x^{\frac{11}{2}} + \frac{10}{3} \, B a^{2} b^{4} x^{\frac{9}{2}} + \frac{4}{3} \, A a b^{5} x^{\frac{9}{2}} + \frac{40}{7} \, B a^{3} b^{3} x^{\frac{7}{2}} + \frac{30}{7} \, A a^{2} b^{4} x^{\frac{7}{2}} + 6 \, B a^{4} b^{2} x^{\frac{5}{2}} + 8 \, A a^{3} b^{3} x^{\frac{5}{2}} + 4 \, B a^{5} b x^{\frac{3}{2}} + 10 \, A a^{4} b^{2} x^{\frac{3}{2}} + 2 \, B a^{6} \sqrt{x} + 12 \, A a^{5} b \sqrt{x} - \frac{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^(3/2),x, algorithm="giac")

[Out]

2/13*B*b^6*x^(13/2) + 12/11*B*a*b^5*x^(11/2) + 2/11*A*b^6*x^(11/2) + 10/3*B*a^2*b^4*x^(9/2) + 4/3*A*a*b^5*x^(9
/2) + 40/7*B*a^3*b^3*x^(7/2) + 30/7*A*a^2*b^4*x^(7/2) + 6*B*a^4*b^2*x^(5/2) + 8*A*a^3*b^3*x^(5/2) + 4*B*a^5*b*
x^(3/2) + 10*A*a^4*b^2*x^(3/2) + 2*B*a^6*sqrt(x) + 12*A*a^5*b*sqrt(x) - 2*A*a^6/sqrt(x)