3.342 \(\int \frac{(a+b x)^3 (A+B x)}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.035442, 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} \[ -\frac{2 a^2 (a B+3 A b)}{3 x^{3/2}}-\frac{2 a^3 A}{5 x^{5/2}}+2 b^2 \sqrt{x} (3 a B+A b)-\frac{6 a b (a B+A b)}{\sqrt{x}}+\frac{2}{3} b^3 B x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^(7/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0206252, size = 65, normalized size = 0.8 \[ -\frac{2 \left (15 a^2 b x (A+3 B x)+a^3 (3 A+5 B x)+45 a b^2 x^2 (A-B x)-5 b^3 x^3 (3 A+B x)\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 76, normalized size = 0.9 \begin{align*} -{\frac{-10\,B{b}^{3}{x}^{4}-30\,A{b}^{3}{x}^{3}-90\,B{x}^{3}a{b}^{2}+90\,aA{b}^{2}{x}^{2}+90\,B{x}^{2}{a}^{2}b+30\,{a}^{2}Abx+10\,{a}^{3}Bx+6\,{a}^{3}A}{15}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.04413, size = 100, normalized size = 1.23 \begin{align*} \frac{2}{3} \, B b^{3} x^{\frac{3}{2}} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.37043, size = 165, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (5 \, B b^{3} x^{4} - 3 \, A a^{3} + 15 \,{\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} - 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.89761, size = 105, normalized size = 1.3 \begin{align*} - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} - \frac{2 A a^{2} b}{x^{\frac{3}{2}}} - \frac{6 A a b^{2}}{\sqrt{x}} + 2 A b^{3} \sqrt{x} - \frac{2 B a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 B a^{2} b}{\sqrt{x}} + 6 B a b^{2} \sqrt{x} + \frac{2 B b^{3} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**(7/2),x)

[Out]

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

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Giac [A]  time = 1.14923, size = 103, normalized size = 1.27 \begin{align*} \frac{2}{3} \, B b^{3} x^{\frac{3}{2}} + 6 \, B a b^{2} \sqrt{x} + 2 \, A b^{3} \sqrt{x} - \frac{2 \,{\left (45 \, B a^{2} b x^{2} + 45 \, A a b^{2} x^{2} + 5 \, B a^{3} x + 15 \, A a^{2} b x + 3 \, A a^{3}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/x^(7/2),x, algorithm="giac")

[Out]

2/3*B*b^3*x^(3/2) + 6*B*a*b^2*sqrt(x) + 2*A*b^3*sqrt(x) - 2/15*(45*B*a^2*b*x^2 + 45*A*a*b^2*x^2 + 5*B*a^3*x +
15*A*a^2*b*x + 3*A*a^3)/x^(5/2)