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

Optimal. Leaf size=59 \[ -\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a (a B+2 A b)}{3 x^{3/2}}-\frac{2 b (2 a B+A b)}{\sqrt{x}}+2 b^2 B \sqrt{x} \]

[Out]

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

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Rubi [A]  time = 0.0250903, antiderivative size = 59, 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} \[ -\frac{2 a^2 A}{5 x^{5/2}}-\frac{2 a (a B+2 A b)}{3 x^{3/2}}-\frac{2 b (2 a B+A b)}{\sqrt{x}}+2 b^2 B \sqrt{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0151995, size = 47, normalized size = 0.8 \[ -\frac{2 \left (a^2 (3 A+5 B x)+10 a b x (A+3 B x)+15 b^2 x^2 (A-B x)\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(15*b^2*x^2*(A - B*x) + 10*a*b*x*(A + 3*B*x) + a^2*(3*A + 5*B*x)))/(15*x^(5/2))

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Maple [A]  time = 0.005, size = 52, normalized size = 0.9 \begin{align*} -{\frac{-30\,B{b}^{2}{x}^{3}+30\,A{b}^{2}{x}^{2}+60\,B{x}^{2}ab+20\,aAbx+10\,{a}^{2}Bx+6\,{a}^{2}A}{15}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-15*B*b^2*x^3+15*A*b^2*x^2+30*B*a*b*x^2+10*A*a*b*x+5*B*a^2*x+3*A*a^2)/x^(5/2)

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Maxima [A]  time = 1.0908, size = 70, normalized size = 1.19 \begin{align*} 2 \, B b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 5 \,{\left (B a^{2} + 2 \, A a 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)^2*(B*x+A)/x^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.36147, size = 122, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (15 \, B b^{2} x^{3} - 3 \, A a^{2} - 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} - 5 \,{\left (B a^{2} + 2 \, A a 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)^2*(B*x+A)/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.1037, size = 75, normalized size = 1.27 \begin{align*} - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b}{3 x^{\frac{3}{2}}} - \frac{2 A b^{2}}{\sqrt{x}} - \frac{2 B a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 B a b}{\sqrt{x}} + 2 B b^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18931, size = 70, normalized size = 1.19 \begin{align*} 2 \, B b^{2} \sqrt{x} - \frac{2 \,{\left (30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 5 \, B a^{2} x + 10 \, A a b x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*b^2*sqrt(x) - 2/15*(30*B*a*b*x^2 + 15*A*b^2*x^2 + 5*B*a^2*x + 10*A*a*b*x + 3*A*a^2)/x^(5/2)