3.542 \(\int \frac {A+B x}{x^{3/2} (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac {4 \sqrt {x} (4 A b-a B)}{3 a^3 \sqrt {a+b x}}-\frac {2 \sqrt {x} (4 A b-a B)}{3 a^2 (a+b x)^{3/2}}-\frac {2 A}{a \sqrt {x} (a+b x)^{3/2}} \]

[Out]

-2*A/a/(b*x+a)^(3/2)/x^(1/2)-2/3*(4*A*b-B*a)*x^(1/2)/a^2/(b*x+a)^(3/2)-4/3*(4*A*b-B*a)*x^(1/2)/a^3/(b*x+a)^(1/
2)

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Rubi [A]  time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ -\frac {4 \sqrt {x} (4 A b-a B)}{3 a^3 \sqrt {a+b x}}-\frac {2 \sqrt {x} (4 A b-a B)}{3 a^2 (a+b x)^{3/2}}-\frac {2 A}{a \sqrt {x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(a*Sqrt[x]*(a + b*x)^(3/2)) - (2*(4*A*b - a*B)*Sqrt[x])/(3*a^2*(a + b*x)^(3/2)) - (4*(4*A*b - a*B)*Sqrt
[x])/(3*a^3*Sqrt[a + b*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{3/2} (a+b x)^{5/2}} \, dx &=-\frac {2 A}{a \sqrt {x} (a+b x)^{3/2}}+\frac {\left (2 \left (-2 A b+\frac {a B}{2}\right )\right ) \int \frac {1}{\sqrt {x} (a+b x)^{5/2}} \, dx}{a}\\ &=-\frac {2 A}{a \sqrt {x} (a+b x)^{3/2}}-\frac {2 (4 A b-a B) \sqrt {x}}{3 a^2 (a+b x)^{3/2}}-\frac {(2 (4 A b-a B)) \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx}{3 a^2}\\ &=-\frac {2 A}{a \sqrt {x} (a+b x)^{3/2}}-\frac {2 (4 A b-a B) \sqrt {x}}{3 a^2 (a+b x)^{3/2}}-\frac {4 (4 A b-a B) \sqrt {x}}{3 a^3 \sqrt {a+b x}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 54, normalized size = 0.67 \[ \frac {-6 a^2 (A-B x)+4 a b x (B x-6 A)-16 A b^2 x^2}{3 a^3 \sqrt {x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*A*b^2*x^2 - 6*a^2*(A - B*x) + 4*a*b*x*(-6*A + B*x))/(3*a^3*Sqrt[x]*(a + b*x)^(3/2))

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fricas [A]  time = 0.67, size = 76, normalized size = 0.94 \[ -\frac {2 \, {\left (3 \, A a^{2} - 2 \, {\left (B a b - 4 \, A b^{2}\right )} x^{2} - 3 \, {\left (B a^{2} - 4 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.79, size = 211, normalized size = 2.60 \[ -\frac {2 \, \sqrt {b x + a} A b^{2}}{\sqrt {{\left (b x + a\right )} b - a b} a^{3} {\left | b \right |}} + \frac {4 \, {\left (6 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} - 3 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {5}{2}} + 2 \, B a^{3} b^{\frac {7}{2}} - 12 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {7}{2}} - 5 \, A a^{2} b^{\frac {9}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{2} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

-2*sqrt(b*x + a)*A*b^2/(sqrt((b*x + a)*b - a*b)*a^3*abs(b)) + 4/3*(6*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x
+ a)*b - a*b))^2*b^(5/2) - 3*A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(5/2) + 2*B*a^3*b^(7/2) -
12*A*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(7/2) - 5*A*a^2*b^(9/2))/(((sqrt(b*x + a)*sqrt(b
) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^2*abs(b))

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maple [A]  time = 0.00, size = 53, normalized size = 0.65 \[ -\frac {2 \left (8 A \,b^{2} x^{2}-2 B a b \,x^{2}+12 A a b x -3 B \,a^{2} x +3 A \,a^{2}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} a^{3} \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^(5/2)/x^(3/2),x)

[Out]

-2/3*(8*A*b^2*x^2-2*B*a*b*x^2+12*A*a*b*x-3*B*a^2*x+3*A*a^2)/(b*x+a)^(3/2)/x^(1/2)/a^3

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maxima [B]  time = 1.02, size = 151, normalized size = 1.86 \[ -\frac {2 \, B a}{3 \, {\left (\sqrt {b x^{2} + a x} a b^{2} x + \sqrt {b x^{2} + a x} a^{2} b\right )}} + \frac {2 \, A}{3 \, {\left (\sqrt {b x^{2} + a x} a b x + \sqrt {b x^{2} + a x} a^{2}\right )}} + \frac {4 \, B x}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {16 \, A b x}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {8 \, A}{3 \, \sqrt {b x^{2} + a x} a^{2}} + \frac {2 \, B}{3 \, \sqrt {b x^{2} + a x} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

-2/3*B*a/(sqrt(b*x^2 + a*x)*a*b^2*x + sqrt(b*x^2 + a*x)*a^2*b) + 2/3*A/(sqrt(b*x^2 + a*x)*a*b*x + sqrt(b*x^2 +
a*x)*a^2) + 4/3*B*x/(sqrt(b*x^2 + a*x)*a^2) - 16/3*A*b*x/(sqrt(b*x^2 + a*x)*a^3) - 8/3*A/(sqrt(b*x^2 + a*x)*a
^2) + 2/3*B/(sqrt(b*x^2 + a*x)*a*b)

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mupad [B]  time = 0.91, size = 88, normalized size = 1.09 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{a\,b^2}-\frac {x\,\left (6\,B\,a^2-24\,A\,a\,b\right )}{3\,a^3\,b^2}+\frac {x^2\,\left (16\,A\,b^2-4\,B\,a\,b\right )}{3\,a^3\,b^2}\right )}{x^{5/2}+\frac {2\,a\,x^{3/2}}{b}+\frac {a^2\,\sqrt {x}}{b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(3/2)*(a + b*x)^(5/2)),x)

[Out]

-((a + b*x)^(1/2)*((2*A)/(a*b^2) - (x*(6*B*a^2 - 24*A*a*b))/(3*a^3*b^2) + (x^2*(16*A*b^2 - 4*B*a*b))/(3*a^3*b^
2)))/(x^(5/2) + (2*a*x^(3/2))/b + (a^2*x^(1/2))/b^2)

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sympy [B]  time = 156.23, size = 250, normalized size = 3.09 \[ A \left (- \frac {6 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {24 a b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {16 b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}}\right ) + B \left (\frac {6 a}{3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} + 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} + 1}} + \frac {4 b x}{3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} + 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(3/2)/(b*x+a)**(5/2),x)

[Out]

A*(-6*a**2*b**(9/2)*sqrt(a/(b*x) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x + 3*a**3*b**6*x**2) - 24*a*b**(11/2)*x*sqrt
(a/(b*x) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x + 3*a**3*b**6*x**2) - 16*b**(13/2)*x**2*sqrt(a/(b*x) + 1)/(3*a**5*b
**4 + 6*a**4*b**5*x + 3*a**3*b**6*x**2)) + B*(6*a/(3*a**3*sqrt(b)*sqrt(a/(b*x) + 1) + 3*a**2*b**(3/2)*x*sqrt(a
/(b*x) + 1)) + 4*b*x/(3*a**3*sqrt(b)*sqrt(a/(b*x) + 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) + 1)))

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