3.6.43 \(\int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx\) [543]

Optimal. Leaf size=113 \[ -\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}+\frac {16 (2 A b-a B) \sqrt {a+b x}}{3 a^4 \sqrt {x}} \]

[Out]

-2/3*A/a/x^(3/2)/(b*x+a)^(3/2)-2/3*(2*A*b-B*a)/a^2/(b*x+a)^(3/2)/x^(1/2)-8/3*(2*A*b-B*a)/a^3/x^(1/2)/(b*x+a)^(
1/2)+16/3*(2*A*b-B*a)*(b*x+a)^(1/2)/a^4/x^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \begin {gather*} \frac {16 \sqrt {a+b x} (2 A b-a B)}{3 a^4 \sqrt {x}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A)/(3*a*x^(3/2)*(a + b*x)^(3/2)) - (2*(2*A*b - a*B))/(3*a^2*Sqrt[x]*(a + b*x)^(3/2)) - (8*(2*A*b - a*B))/(
3*a^3*Sqrt[x]*Sqrt[a + b*x]) + (16*(2*A*b - a*B)*Sqrt[a + b*x])/(3*a^4*Sqrt[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 47

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 79

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] || L
tQ[p, n]))))

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {\left (2 \left (-3 A b+\frac {3 a B}{2}\right )\right ) \int \frac {1}{x^{3/2} (a+b x)^{5/2}} \, dx}{3 a}\\ &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {(4 (2 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx}{3 a^2}\\ &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}-\frac {(8 (2 A b-a B)) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^3}\\ &=-\frac {2 A}{3 a x^{3/2} (a+b x)^{3/2}}-\frac {2 (2 A b-a B)}{3 a^2 \sqrt {x} (a+b x)^{3/2}}-\frac {8 (2 A b-a B)}{3 a^3 \sqrt {x} \sqrt {a+b x}}+\frac {16 (2 A b-a B) \sqrt {a+b x}}{3 a^4 \sqrt {x}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 70, normalized size = 0.62 \begin {gather*} -\frac {2 \left (-16 A b^3 x^3-6 a^2 b x (A-2 B x)+8 a b^2 x^2 (-3 A+B x)+a^3 (A+3 B x)\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-16*A*b^3*x^3 - 6*a^2*b*x*(A - 2*B*x) + 8*a*b^2*x^2*(-3*A + B*x) + a^3*(A + 3*B*x)))/(3*a^4*x^(3/2)*(a +
b*x)^(3/2))

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Maple [A]
time = 0.09, size = 76, normalized size = 0.67

method	result	size
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-8 A b x +3 B a x +A a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {2 b \left (8 b^{2} A x -5 a b B x +9 a b A -6 a^{2} B \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{4}}\)	\(72\)
gosper	\(-\frac {2 \left (-16 A \,b^{3} x^{3}+8 B a \,b^{2} x^{3}-24 a A \,b^{2} x^{2}+12 B \,a^{2} b \,x^{2}-6 a^{2} A b x +3 a^{3} B x +a^{3} A \right )}{3 \left (b x +a \right )^{\frac {3}{2}} x^{\frac {3}{2}} a^{4}}\)	\(76\)
default	\(-\frac {2 \left (-16 A \,b^{3} x^{3}+8 B a \,b^{2} x^{3}-24 a A \,b^{2} x^{2}+12 B \,a^{2} b \,x^{2}-6 a^{2} A b x +3 a^{3} B x +a^{3} A \right )}{3 \left (b x +a \right )^{\frac {3}{2}} x^{\frac {3}{2}} a^{4}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-16*A*b^3*x^3+8*B*a*b^2*x^3-24*A*a*b^2*x^2+12*B*a^2*b*x^2-6*A*a^2*b*x+3*B*a^3*x+A*a^3)/(b*x+a)^(3/2)/x^(
3/2)/a^4

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Maxima [A]
time = 0.28, size = 130, normalized size = 1.15 \begin {gather*} \frac {2 \, B x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} - \frac {16 \, B b x}{3 \, \sqrt {b x^{2} + a x} a^{3}} - \frac {4 \, A b x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} + \frac {32 \, A b^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {8 \, B}{3 \, \sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, A}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} + \frac {16 \, A b}{3 \, \sqrt {b x^{2} + a x} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.99, size = 100, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (A a^{3} + 8 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 12 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(A*a^3 + 8*(B*a*b^2 - 2*A*b^3)*x^3 + 12*(B*a^2*b - 2*A*a*b^2)*x^2 + 3*(B*a^3 - 2*A*a^2*b)*x)*sqrt(b*x + a
)*sqrt(x)/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (110) = 220\).
time = 194.60, size = 495, normalized size = 4.38 \begin {gather*} A \left (- \frac {2 a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 a^{3} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {60 a^{2} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 a b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {32 b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}}\right ) + B \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(-2*a**4*b**(19/2)*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*x**3 + 3*a**4*b**12*x
**4) + 10*a**3*b**(21/2)*x*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*x**3 + 3*a**4*b
**12*x**4) + 60*a**2*b**(23/2)*x**2*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*x**3 +
3*a**4*b**12*x**4) + 80*a*b**(25/2)*x**3*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b**11*
x**3 + 3*a**4*b**12*x**4) + 32*b**(27/2)*x**4*sqrt(a/(b*x) + 1)/(3*a**7*b**9*x + 9*a**6*b**10*x**2 + 9*a**5*b*
*11*x**3 + 3*a**4*b**12*x**4)) + B*(-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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (86) = 172\).
time = 1.13, size = 303, normalized size = 2.68 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (3 \, B a^{4} b^{3} {\left | b \right |} - 8 \, A a^{3} b^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{7} b^{2}} - \frac {3 \, {\left (B a^{5} b^{3} {\left | b \right |} - 3 \, A a^{4} b^{4} {\left | b \right |}\right )}}{a^{7} b^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (3 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {5}{2}} + 12 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {7}{2}} - 6 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {7}{2}} + 5 \, B a^{3} b^{\frac {9}{2}} - 18 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} - 8 \, A a^{2} b^{\frac {11}{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^{3} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b*x + a)*((3*B*a^4*b^3*abs(b) - 8*A*a^3*b^4*abs(b))*(b*x + a)/(a^7*b^2) - 3*(B*a^5*b^3*abs(b) - 3*A*
a^4*b^4*abs(b))/(a^7*b^2))/((b*x + a)*b - a*b)^(3/2) - 4/3*(3*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b -
a*b))^4*b^(5/2) + 12*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(7/2) - 6*A*(sqrt(b*x + a)*sq
rt(b) - sqrt((b*x + a)*b - a*b))^4*b^(7/2) + 5*B*a^3*b^(9/2) - 18*A*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*
b - a*b))^2*b^(9/2) - 8*A*a^2*b^(11/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^3*abs
(b))

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Mupad [B]
time = 0.97, size = 112, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{3\,a\,b^2}-\frac {8\,x^2\,\left (2\,A\,b-B\,a\right )}{a^3\,b}-\frac {x^3\,\left (32\,A\,b^3-16\,B\,a\,b^2\right )}{3\,a^4\,b^2}+\frac {x\,\left (6\,B\,a^3-12\,A\,a^2\,b\right )}{3\,a^4\,b^2}\right )}{x^{7/2}+\frac {2\,a\,x^{5/2}}{b}+\frac {a^2\,x^{3/2}}{b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a + b*x)^(1/2)*((2*A)/(3*a*b^2) - (8*x^2*(2*A*b - B*a))/(a^3*b) - (x^3*(32*A*b^3 - 16*B*a*b^2))/(3*a^4*b^2)
+ (x*(6*B*a^3 - 12*A*a^2*b))/(3*a^4*b^2)))/(x^(7/2) + (2*a*x^(5/2))/b + (a^2*x^(3/2))/b^2)

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