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

Optimal. Leaf size=107 \[ \frac{4}{3} a^2 b x^{3/2} (2 a B+3 A b)+2 a^3 \sqrt{x} (a B+4 A b)-\frac{2 a^4 A}{\sqrt{x}}+\frac{2}{7} b^3 x^{7/2} (4 a B+A b)+\frac{4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac{2}{9} b^4 B x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.0510342, antiderivative size = 107, 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} \[ \frac{4}{3} a^2 b x^{3/2} (2 a B+3 A b)+2 a^3 \sqrt{x} (a B+4 A b)-\frac{2 a^4 A}{\sqrt{x}}+\frac{2}{7} b^3 x^{7/2} (4 a B+A b)+\frac{4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac{2}{9} b^4 B x^{9/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0245927, size = 87, normalized size = 0.81 \[ \frac{252 a^2 b^2 x^2 (5 A+3 B x)+840 a^3 b x (3 A+B x)-630 a^4 (A-B x)+72 a b^3 x^3 (7 A+5 B x)+10 b^4 x^4 (9 A+7 B x)}{315 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-630*a^4*(A - B*x) + 840*a^3*b*x*(3*A + B*x) + 252*a^2*b^2*x^2*(5*A + 3*B*x) + 72*a*b^3*x^3*(7*A + 5*B*x) + 1
0*b^4*x^4*(9*A + 7*B*x))/(315*Sqrt[x])

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Maple [A]  time = 0.009, size = 100, normalized size = 0.9 \begin{align*} -{\frac{-70\,{b}^{4}B{x}^{5}-90\,A{b}^{4}{x}^{4}-360\,B{x}^{4}a{b}^{3}-504\,aA{b}^{3}{x}^{3}-756\,B{x}^{3}{a}^{2}{b}^{2}-1260\,{a}^{2}A{b}^{2}{x}^{2}-840\,B{x}^{2}{a}^{3}b-2520\,{a}^{3}Abx-630\,{a}^{4}Bx+630\,A{a}^{4}}{315}{\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)^2/x^(3/2),x)

[Out]

-2/315*(-35*B*b^4*x^5-45*A*b^4*x^4-180*B*a*b^3*x^4-252*A*a*b^3*x^3-378*B*a^2*b^2*x^3-630*A*a^2*b^2*x^2-420*B*a
^3*b*x^2-1260*A*a^3*b*x-315*B*a^4*x+315*A*a^4)/x^(1/2)

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Maxima [A]  time = 0.999749, size = 134, normalized size = 1.25 \begin{align*} \frac{2}{9} \, B b^{4} x^{\frac{9}{2}} - \frac{2 \, A a^{4}}{\sqrt{x}} + \frac{2}{7} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{7}{2}} + \frac{4}{5} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{5}{2}} + \frac{4}{3} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{4} + 4 \, A a^{3} 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)^2/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53138, size = 231, normalized size = 2.16 \begin{align*} \frac{2 \,{\left (35 \, B b^{4} x^{5} - 315 \, A a^{4} + 45 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 126 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 210 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 315 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{315 \, \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)^2/x^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^4*x^5 - 315*A*a^4 + 45*(4*B*a*b^3 + A*b^4)*x^4 + 126*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 210*(2*B*a^
3*b + 3*A*a^2*b^2)*x^2 + 315*(B*a^4 + 4*A*a^3*b)*x)/sqrt(x)

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Sympy [A]  time = 2.0922, size = 141, normalized size = 1.32 \begin{align*} - \frac{2 A a^{4}}{\sqrt{x}} + 8 A a^{3} b \sqrt{x} + 4 A a^{2} b^{2} x^{\frac{3}{2}} + \frac{8 A a b^{3} x^{\frac{5}{2}}}{5} + \frac{2 A b^{4} x^{\frac{7}{2}}}{7} + 2 B a^{4} \sqrt{x} + \frac{8 B a^{3} b x^{\frac{3}{2}}}{3} + \frac{12 B a^{2} b^{2} x^{\frac{5}{2}}}{5} + \frac{8 B a b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{4} x^{\frac{9}{2}}}{9} \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)**2/x**(3/2),x)

[Out]

-2*A*a**4/sqrt(x) + 8*A*a**3*b*sqrt(x) + 4*A*a**2*b**2*x**(3/2) + 8*A*a*b**3*x**(5/2)/5 + 2*A*b**4*x**(7/2)/7
+ 2*B*a**4*sqrt(x) + 8*B*a**3*b*x**(3/2)/3 + 12*B*a**2*b**2*x**(5/2)/5 + 8*B*a*b**3*x**(7/2)/7 + 2*B*b**4*x**(
9/2)/9

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Giac [A]  time = 1.18998, size = 136, normalized size = 1.27 \begin{align*} \frac{2}{9} \, B b^{4} x^{\frac{9}{2}} + \frac{8}{7} \, B a b^{3} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{4} x^{\frac{7}{2}} + \frac{12}{5} \, B a^{2} b^{2} x^{\frac{5}{2}} + \frac{8}{5} \, A a b^{3} x^{\frac{5}{2}} + \frac{8}{3} \, B a^{3} b x^{\frac{3}{2}} + 4 \, A a^{2} b^{2} x^{\frac{3}{2}} + 2 \, B a^{4} \sqrt{x} + 8 \, A a^{3} b \sqrt{x} - \frac{2 \, A a^{4}}{\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)^2/x^(3/2),x, algorithm="giac")

[Out]

2/9*B*b^4*x^(9/2) + 8/7*B*a*b^3*x^(7/2) + 2/7*A*b^4*x^(7/2) + 12/5*B*a^2*b^2*x^(5/2) + 8/5*A*a*b^3*x^(5/2) + 8
/3*B*a^3*b*x^(3/2) + 4*A*a^2*b^2*x^(3/2) + 2*B*a^4*sqrt(x) + 8*A*a^3*b*sqrt(x) - 2*A*a^4/sqrt(x)