3.341 \(\int \frac{(a+b x)^3 (A+B x)}{x^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt{x} (a B+A b)+\frac{2}{5} b^3 B x^{5/2} \]

[Out]

(-2*a^3*A)/(3*x^(3/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 6*a*b*(A*b + a*B)*Sqrt[x] + (2*b^2*(A*b + 3*a*B)*x^(3
/2))/3 + (2*b^3*B*x^(5/2))/5

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Rubi [A]  time = 0.035759, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {76} \[ -\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt{x} (a B+A b)+\frac{2}{5} b^3 B x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^(5/2),x]

[Out]

(-2*a^3*A)/(3*x^(3/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 6*a*b*(A*b + a*B)*Sqrt[x] + (2*b^2*(A*b + 3*a*B)*x^(3
/2))/3 + (2*b^3*B*x^(5/2))/5

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

Mathematica [A]  time = 0.0207379, size = 66, normalized size = 0.81 \[ \frac{2 \left (45 a^2 b x (B x-A)-5 a^3 (A+3 B x)+15 a b^2 x^2 (3 A+B x)+b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/x^(5/2),x]

[Out]

(2*(45*a^2*b*x*(-A + B*x) + 15*a*b^2*x^2*(3*A + B*x) - 5*a^3*(A + 3*B*x) + b^3*x^3*(5*A + 3*B*x)))/(15*x^(3/2)
)

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Maple [A]  time = 0.006, size = 76, normalized size = 0.9 \begin{align*} -{\frac{-6\,B{b}^{3}{x}^{4}-10\,A{b}^{3}{x}^{3}-30\,B{x}^{3}a{b}^{2}-90\,aA{b}^{2}{x}^{2}-90\,B{x}^{2}{a}^{2}b+90\,{a}^{2}Abx+30\,{a}^{3}Bx+10\,{a}^{3}A}{15}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^(5/2),x)

[Out]

-2/15*(-3*B*b^3*x^4-5*A*b^3*x^3-15*B*a*b^2*x^3-45*A*a*b^2*x^2-45*B*a^2*b*x^2+45*A*a^2*b*x+15*B*a^3*x+5*A*a^3)/
x^(3/2)

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Maxima [A]  time = 1.08934, size = 99, normalized size = 1.22 \begin{align*} \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{3}{2}} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A a^{3} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^(5/2),x, algorithm="maxima")

[Out]

2/5*B*b^3*x^(5/2) + 2/3*(3*B*a*b^2 + A*b^3)*x^(3/2) + 6*(B*a^2*b + A*a*b^2)*sqrt(x) - 2/3*(A*a^3 + 3*(B*a^3 +
3*A*a^2*b)*x)/x^(3/2)

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Fricas [A]  time = 2.28619, size = 165, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^3*x^4 - 5*A*a^3 + 5*(3*B*a*b^2 + A*b^3)*x^3 + 45*(B*a^2*b + A*a*b^2)*x^2 - 15*(B*a^3 + 3*A*a^2*b)*
x)/x^(3/2)

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Sympy [A]  time = 1.69225, size = 105, normalized size = 1.3 \begin{align*} - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 A a^{2} b}{\sqrt{x}} + 6 A a b^{2} \sqrt{x} + \frac{2 A b^{3} x^{\frac{3}{2}}}{3} - \frac{2 B a^{3}}{\sqrt{x}} + 6 B a^{2} b \sqrt{x} + 2 B a b^{2} x^{\frac{3}{2}} + \frac{2 B b^{3} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**(5/2),x)

[Out]

-2*A*a**3/(3*x**(3/2)) - 6*A*a**2*b/sqrt(x) + 6*A*a*b**2*sqrt(x) + 2*A*b**3*x**(3/2)/3 - 2*B*a**3/sqrt(x) + 6*
B*a**2*b*sqrt(x) + 2*B*a*b**2*x**(3/2) + 2*B*b**3*x**(5/2)/5

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Giac [A]  time = 1.34927, size = 101, normalized size = 1.25 \begin{align*} \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + 2 \, B a b^{2} x^{\frac{3}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} + 6 \, B a^{2} b \sqrt{x} + 6 \, A a b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B a^{3} x + 9 \, A a^{2} b x + A a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^(5/2),x, algorithm="giac")

[Out]

2/5*B*b^3*x^(5/2) + 2*B*a*b^2*x^(3/2) + 2/3*A*b^3*x^(3/2) + 6*B*a^2*b*sqrt(x) + 6*A*a*b^2*sqrt(x) - 2/3*(3*B*a
^3*x + 9*A*a^2*b*x + A*a^3)/x^(3/2)