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

Optimal. Leaf size=35 \[ -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 b B \sqrt {x} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 b B \sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/x^(5/2),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.80 \begin {gather*} -\frac {2 (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 31, normalized size = 0.89 \begin {gather*} \frac {2 \left (-a A-3 a B x-3 A b x+3 b B x^2\right )}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.06, size = 27, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (3 \, B b x^{2} - A a - 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.20, size = 27, normalized size = 0.77 \begin {gather*} 2 \, B b \sqrt {x} - \frac {2 \, {\left (3 \, B a x + 3 \, A b x + A a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 27, normalized size = 0.77 \begin {gather*} -\frac {2 \left (-3 B b \,x^{2}+3 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.90, size = 27, normalized size = 0.77 \begin {gather*} 2 \, B b \sqrt {x} - \frac {2 \, {\left (A a + 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.34, size = 27, normalized size = 0.77 \begin {gather*} -\frac {2\,A\,a+6\,A\,b\,x+6\,B\,a\,x-6\,B\,b\,x^2}{3\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x))/x^(5/2),x)

[Out]

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

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sympy [A]  time = 0.64, size = 41, normalized size = 1.17 \begin {gather*} - \frac {2 A a}{3 x^{\frac {3}{2}}} - \frac {2 A b}{\sqrt {x}} - \frac {2 B a}{\sqrt {x}} + 2 B b \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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