\(\int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx\) [487]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}-\frac {8 b (2 A b-3 a B) (a+b x)^{3/2}}{105 a^3 x^{5/2}}+\frac {16 b^2 (2 A b-3 a B) (a+b x)^{3/2}}{315 a^4 x^{3/2}} \]

[Out]

-2/9*A*(b*x+a)^(3/2)/a/x^(9/2)+2/21*(2*A*b-3*B*a)*(b*x+a)^(3/2)/a^2/x^(7/2)-8/105*b*(2*A*b-3*B*a)*(b*x+a)^(3/2
)/a^3/x^(5/2)+16/315*b^2*(2*A*b-3*B*a)*(b*x+a)^(3/2)/a^4/x^(3/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=\frac {16 b^2 (a+b x)^{3/2} (2 A b-3 a B)}{315 a^4 x^{3/2}}-\frac {8 b (a+b x)^{3/2} (2 A b-3 a B)}{105 a^3 x^{5/2}}+\frac {2 (a+b x)^{3/2} (2 A b-3 a B)}{21 a^2 x^{7/2}}-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}} \]

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/x^(11/2),x]

[Out]

(-2*A*(a + b*x)^(3/2))/(9*a*x^(9/2)) + (2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(21*a^2*x^(7/2)) - (8*b*(2*A*b - 3*
a*B)*(a + b*x)^(3/2))/(105*a^3*x^(5/2)) + (16*b^2*(2*A*b - 3*a*B)*(a + b*x)^(3/2))/(315*a^4*x^(3/2))

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*} \text {integral}& = -\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {\left (2 \left (-3 A b+\frac {9 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{x^{9/2}} \, dx}{9 a} \\ & = -\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}+\frac {(4 b (2 A b-3 a B)) \int \frac {\sqrt {a+b x}}{x^{7/2}} \, dx}{21 a^2} \\ & = -\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}-\frac {8 b (2 A b-3 a B) (a+b x)^{3/2}}{105 a^3 x^{5/2}}-\frac {\left (8 b^2 (2 A b-3 a B)\right ) \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx}{105 a^3} \\ & = -\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}-\frac {8 b (2 A b-3 a B) (a+b x)^{3/2}}{105 a^3 x^{5/2}}+\frac {16 b^2 (2 A b-3 a B) (a+b x)^{3/2}}{315 a^4 x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=-\frac {2 (a+b x)^{3/2} \left (-16 A b^3 x^3+24 a b^2 x^2 (A+B x)-6 a^2 b x (5 A+6 B x)+5 a^3 (7 A+9 B x)\right )}{315 a^4 x^{9/2}} \]

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(11/2),x]

[Out]

(-2*(a + b*x)^(3/2)*(-16*A*b^3*x^3 + 24*a*b^2*x^2*(A + B*x) - 6*a^2*b*x*(5*A + 6*B*x) + 5*a^3*(7*A + 9*B*x)))/
(315*a^4*x^(9/2))

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} x^{3}+24 B a \,b^{2} x^{3}+24 a A \,b^{2} x^{2}-36 B \,a^{2} b \,x^{2}-30 a^{2} A b x +45 a^{3} B x +35 a^{3} A \right )}{315 x^{\frac {9}{2}} a^{4}}\)	\(77\)
default	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} x^{3}+24 B a \,b^{2} x^{3}+24 a A \,b^{2} x^{2}-36 B \,a^{2} b \,x^{2}-30 a^{2} A b x +45 a^{3} B x +35 a^{3} A \right )}{315 x^{\frac {9}{2}} a^{4}}\)	\(77\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-16 A \,b^{4} x^{4}+24 B a \,b^{3} x^{4}+8 A a \,b^{3} x^{3}-12 B \,a^{2} b^{2} x^{3}-6 A \,a^{2} b^{2} x^{2}+9 B \,a^{3} b \,x^{2}+5 A \,a^{3} b x +45 B \,a^{4} x +35 A \,a^{4}\right )}{315 x^{\frac {9}{2}} a^{4}}\)	\(101\)

[In]

int((B*x+A)*(b*x+a)^(1/2)/x^(11/2),x,method=_RETURNVERBOSE)

[Out]

-2/315*(b*x+a)^(3/2)*(-16*A*b^3*x^3+24*B*a*b^2*x^3+24*A*a*b^2*x^2-36*B*a^2*b*x^2-30*A*a^2*b*x+45*B*a^3*x+35*A*
a^3)/x^(9/2)/a^4

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=-\frac {2 \, {\left (35 \, A a^{4} + 8 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} - 4 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 3 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {b x + a}}{315 \, a^{4} x^{\frac {9}{2}}} \]

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(11/2),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^4 + 8*(3*B*a*b^3 - 2*A*b^4)*x^4 - 4*(3*B*a^2*b^2 - 2*A*a*b^3)*x^3 + 3*(3*B*a^3*b - 2*A*a^2*b^2)
*x^2 + 5*(9*B*a^4 + A*a^3*b)*x)*sqrt(b*x + a)/(a^4*x^(9/2))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 930 vs. \(2 (116) = 232\).

Time = 32.81 (sec) , antiderivative size = 930, normalized size of antiderivative = 7.95 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=- \frac {70 A a^{7} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} - \frac {220 A a^{6} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} - \frac {228 A a^{5} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} - \frac {80 A a^{4} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {10 A a^{3} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {60 A a^{2} b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {80 A a b^{\frac {31}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} + \frac {32 A b^{\frac {33}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{315 a^{7} b^{9} x^{4} + 945 a^{6} b^{10} x^{5} + 945 a^{5} b^{11} x^{6} + 315 a^{4} b^{12} x^{7}} - \frac {30 B a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {66 B a^{4} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 B a^{3} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {6 B a^{2} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 B a b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {16 B b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} + 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} \]

[In]

integrate((B*x+A)*(b*x+a)**(1/2)/x**(11/2),x)

[Out]

-70*A*a**7*b**(19/2)*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a**5*b**11*x**6 + 315*a
**4*b**12*x**7) - 220*A*a**6*b**(21/2)*x*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a**
5*b**11*x**6 + 315*a**4*b**12*x**7) - 228*A*a**5*b**(23/2)*x**2*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a*
*6*b**10*x**5 + 945*a**5*b**11*x**6 + 315*a**4*b**12*x**7) - 80*A*a**4*b**(25/2)*x**3*sqrt(a/(b*x) + 1)/(315*a
**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a**5*b**11*x**6 + 315*a**4*b**12*x**7) + 10*A*a**3*b**(27/2)*x**4*sq
rt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a**5*b**11*x**6 + 315*a**4*b**12*x**7) + 60*A*
a**2*b**(29/2)*x**5*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a**5*b**11*x**6 + 315*a*
*4*b**12*x**7) + 80*A*a*b**(31/2)*x**6*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a**5*
b**11*x**6 + 315*a**4*b**12*x**7) + 32*A*b**(33/2)*x**7*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10
*x**5 + 945*a**5*b**11*x**6 + 315*a**4*b**12*x**7) - 30*B*a**5*b**(9/2)*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3
+ 210*a**4*b**5*x**4 + 105*a**3*b**6*x**5) - 66*B*a**4*b**(11/2)*x*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210
*a**4*b**5*x**4 + 105*a**3*b**6*x**5) - 34*B*a**3*b**(13/2)*x**2*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a
**4*b**5*x**4 + 105*a**3*b**6*x**5) - 6*B*a**2*b**(15/2)*x**3*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4
*b**5*x**4 + 105*a**3*b**6*x**5) - 24*B*a*b**(17/2)*x**4*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5
*x**4 + 105*a**3*b**6*x**5) - 16*B*b**(19/2)*x**5*sqrt(a/(b*x) + 1)/(105*a**5*b**4*x**3 + 210*a**4*b**5*x**4 +
105*a**3*b**6*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (93) = 186\).

Time = 0.20 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=-\frac {16 \, \sqrt {b x^{2} + a x} B b^{3}}{105 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{4}}{315 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{3}}{315 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B b}{35 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{2}}{105 \, a^{2} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{7 \, x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{63 \, a x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{9 \, x^{5}} \]

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(11/2),x, algorithm="maxima")

[Out]

-16/105*sqrt(b*x^2 + a*x)*B*b^3/(a^3*x) + 32/315*sqrt(b*x^2 + a*x)*A*b^4/(a^4*x) + 8/105*sqrt(b*x^2 + a*x)*B*b
^2/(a^2*x^2) - 16/315*sqrt(b*x^2 + a*x)*A*b^3/(a^3*x^2) - 2/35*sqrt(b*x^2 + a*x)*B*b/(a*x^3) + 4/105*sqrt(b*x^
2 + a*x)*A*b^2/(a^2*x^3) - 2/7*sqrt(b*x^2 + a*x)*B/x^4 - 2/63*sqrt(b*x^2 + a*x)*A*b/(a*x^4) - 2/9*sqrt(b*x^2 +
a*x)*A/x^5

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=-\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B a b^{8} - 2 \, A b^{9}\right )} {\left (b x + a\right )}}{a^{4}} - \frac {9 \, {\left (3 \, B a^{2} b^{8} - 2 \, A a b^{9}\right )}}{a^{4}}\right )} + \frac {63 \, {\left (3 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )}}{a^{4}}\right )} - \frac {105 \, {\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{4}}\right )} {\left (b x + a\right )}^{\frac {3}{2}} b}{315 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}} {\left | b \right |}} \]

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/x^(11/2),x, algorithm="giac")

[Out]

-2/315*((b*x + a)*(4*(b*x + a)*(2*(3*B*a*b^8 - 2*A*b^9)*(b*x + a)/a^4 - 9*(3*B*a^2*b^8 - 2*A*a*b^9)/a^4) + 63*
(3*B*a^3*b^8 - 2*A*a^2*b^9)/a^4) - 105*(B*a^4*b^8 - A*a^3*b^9)/a^4)*(b*x + a)^(3/2)*b/(((b*x + a)*b - a*b)^(9/
2)*abs(b))

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9}+\frac {x\,\left (90\,B\,a^4+10\,A\,b\,a^3\right )}{315\,a^4}-\frac {x^4\,\left (32\,A\,b^4-48\,B\,a\,b^3\right )}{315\,a^4}+\frac {8\,b^2\,x^3\,\left (2\,A\,b-3\,B\,a\right )}{315\,a^3}-\frac {2\,b\,x^2\,\left (2\,A\,b-3\,B\,a\right )}{105\,a^2}\right )}{x^{9/2}} \]

[In]

int(((A + B*x)*(a + b*x)^(1/2))/x^(11/2),x)

[Out]

-((a + b*x)^(1/2)*((2*A)/9 + (x*(90*B*a^4 + 10*A*a^3*b))/(315*a^4) - (x^4*(32*A*b^4 - 48*B*a*b^3))/(315*a^4) +
(8*b^2*x^3*(2*A*b - 3*B*a))/(315*a^3) - (2*b*x^2*(2*A*b - 3*B*a))/(105*a^2)))/x^(9/2)