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

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

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Rubi [A]  time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 67, normalized size = 0.85 \begin {gather*} \frac {2 \left (-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)\right )}{35 \sqrt {x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.04, size = 79, normalized size = 1.00 \begin {gather*} \frac {2 \left (-35 a^3 A+35 a^3 B x+105 a^2 A b x+35 a^2 b B x^2+35 a A b^2 x^2+21 a b^2 B x^3+7 A b^3 x^3+5 b^3 B x^4\right )}{35 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.86, size = 73, normalized size = 0.92 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.25, size = 77, normalized size = 0.97 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.00, size = 76, normalized size = 0.96 \begin {gather*} -\frac {2 \left (-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^{3} A \right )}{35 \sqrt {x}} \end {gather*}

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 = 0.83, size = 73, normalized size = 0.92 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.04, size = 69, normalized size = 0.87 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{5/2}\,\left (\frac {2\,A\,b^3}{5}+\frac {6\,B\,a\,b^2}{5}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,b^3\,x^{7/2}}{7}+2\,a\,b\,x^{3/2}\,\left (A\,b+B\,a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)*(2*B*a^3 + 6*A*a^2*b) + x^(5/2)*((2*A*b^3)/5 + (6*B*a*b^2)/5) - (2*A*a^3)/x^(1/2) + (2*B*b^3*x^(7/2))/
7 + 2*a*b*x^(3/2)*(A*b + B*a)

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sympy [A]  time = 1.49, size = 105, normalized size = 1.33 \begin {gather*} - \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 {gather*}

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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