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

Optimal. Leaf size=79 \[ 2 a^2 \sqrt{x} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{5} b^2 x^{5/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac{2}{7} b^3 B x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.0384907, antiderivative size = 79, 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} \[ 2 a^2 \sqrt{x} (a B+3 A b)-\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{5} b^2 x^{5/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac{2}{7} b^3 B x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0191451, size = 67, normalized size = 0.85 \[ \frac{2 \left (35 a^2 b x (3 A+B x)-35 a^3 (A-B x)+7 a b^2 x^2 (5 A+3 B x)+b^3 x^3 (7 A+5 B x)\right )}{35 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-35*a^3*(A - B*x) + 35*a^2*b*x*(3*A + B*x) + 7*a*b^2*x^2*(5*A + 3*B*x) + b^3*x^3*(7*A + 5*B*x)))/(35*Sqrt[
x])

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Maple [A]  time = 0.003, size = 76, normalized size = 1. \begin{align*} -{\frac{-10\,B{b}^{3}{x}^{4}-14\,A{b}^{3}{x}^{3}-42\,B{x}^{3}a{b}^{2}-70\,aA{b}^{2}{x}^{2}-70\,B{x}^{2}{a}^{2}b-210\,{a}^{2}Abx-70\,{a}^{3}Bx+70\,{a}^{3}A}{35}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(-5*B*b^3*x^4-7*A*b^3*x^3-21*B*a*b^2*x^3-35*A*a*b^2*x^2-35*B*a^2*b*x^2-105*A*a^2*b*x-35*B*a^3*x+35*A*a^3
)/x^(1/2)

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Maxima [A]  time = 1.07287, size = 99, normalized size = 1.25 \begin{align*} \frac{2}{7} \, B b^{3} x^{\frac{7}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} + \frac{2}{5} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{5}{2}} + 2 \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.39118, size = 166, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (5 \, B b^{3} x^{4} - 35 \, A a^{3} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 35 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*B*b^3*x^4 - 35*A*a^3 + 7*(3*B*a*b^2 + A*b^3)*x^3 + 35*(B*a^2*b + A*a*b^2)*x^2 + 35*(B*a^3 + 3*A*a^2*b)
*x)/sqrt(x)

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Sympy [A]  time = 1.58911, size = 105, normalized size = 1.33 \begin{align*} - \frac{2 A a^{3}}{\sqrt{x}} + 6 A a^{2} b \sqrt{x} + 2 A a b^{2} x^{\frac{3}{2}} + \frac{2 A b^{3} x^{\frac{5}{2}}}{5} + 2 B a^{3} \sqrt{x} + 2 B a^{2} b x^{\frac{3}{2}} + \frac{6 B a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 B b^{3} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**3/sqrt(x) + 6*A*a**2*b*sqrt(x) + 2*A*a*b**2*x**(3/2) + 2*A*b**3*x**(5/2)/5 + 2*B*a**3*sqrt(x) + 2*B*a*
*2*b*x**(3/2) + 6*B*a*b**2*x**(5/2)/5 + 2*B*b**3*x**(7/2)/7

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Giac [A]  time = 1.21482, size = 104, normalized size = 1.32 \begin{align*} \frac{2}{7} \, B b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B a b^{2} x^{\frac{5}{2}} + \frac{2}{5} \, A b^{3} x^{\frac{5}{2}} + 2 \, B a^{2} b x^{\frac{3}{2}} + 2 \, A a b^{2} x^{\frac{3}{2}} + 2 \, B a^{3} \sqrt{x} + 6 \, A a^{2} b \sqrt{x} - \frac{2 \, A a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*B*b^3*x^(7/2) + 6/5*B*a*b^2*x^(5/2) + 2/5*A*b^3*x^(5/2) + 2*B*a^2*b*x^(3/2) + 2*A*a*b^2*x^(3/2) + 2*B*a^3*
sqrt(x) + 6*A*a^2*b*sqrt(x) - 2*A*a^3/sqrt(x)