∫ √ ____ a + bx(A+Bx)______________

Optimal. Leaf size=95 \[ -\frac {2 (B d-A e) (a+b x)^{3/2}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (3 b B d+2 A b e-5 a B e) (a+b x)^{3/2}}{15 e (b d-a e)^2 (d+e x)^{3/2}} \]

-2/5*(-A*e+B*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)/(e*x+d)^(5/2)+2/15*(2*A*b*e-5*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {79, 37} \begin {gather*} \frac {2 (a+b x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \end {gather*}

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(5*e*(b*d - a*e)*(d + e*x)^(5/2)) + (2*(3*b*B*d + 2*A*b*e - 5*a*B*e)*(a + b*x

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

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

\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{7/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {(3 b B d+2 A b e-5 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{5 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{5 e (b d-a e) (d+e x)^{5/2}}+\frac {2 (3 b B d+2 A b e-5 a B e) (a+b x)^{3/2}}{15 e (b d-a e)^2 (d+e x)^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 66, normalized size = 0.69 \begin {gather*} \frac {2 (a+b x)^{3/2} (B (-2 a d+3 b d x-5 a e x)+A (5 b d-3 a e+2 b e x))}{15 (b d-a e)^2 (d+e x)^{5/2}} \end {gather*}

(2*(a + b*x)^(3/2)*(B*(-2*a*d + 3*b*d*x - 5*a*e*x) + A*(5*b*d - 3*a*e + 2*b*e*x)))/(15*(b*d - a*e)^2*(d + e*x)

________________________________________________________________________________________


method	result	size

default	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-2 A b e x +5 B a e x -3 B b d x +3 A a e -5 A b d +2 B a d \right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{2}}\)	\(61\)
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-2 A b e x +5 B a e x -3 B b d x +3 A a e -5 A b d +2 B a d \right )}{15 \left (e x +d \right )^{\frac {5}{2}} \left (a^{2} e^{2}-2 b e a d +b^{2} d^{2}\right )}\)	\(74\)

-2/15*(b*x+a)^(3/2)/(e*x+d)^(5/2)*(-2*A*b*e*x+5*B*a*e*x-3*B*b*d*x+3*A*a*e-5*A*b*d+2*B*a*d)/(a*e-b*d)^2

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (88) = 176\).
time = 4.80, size = 221, normalized size = 2.33 \begin {gather*} \frac {2 \, {\left (3 \, B b^{2} d x^{2} + {\left (B a b + 5 \, A b^{2}\right )} d x - {\left (2 \, B a^{2} - 5 \, A a b\right )} d - {\left (3 \, A a^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} x^{2} + {\left (5 \, B a^{2} + A a b\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{15 \, {\left (b^{2} d^{5} + a^{2} x^{3} e^{5} - {\left (2 \, a b d x^{3} - 3 \, a^{2} d x^{2}\right )} e^{4} + {\left (b^{2} d^{2} x^{3} - 6 \, a b d^{2} x^{2} + 3 \, a^{2} d^{2} x\right )} e^{3} + {\left (3 \, b^{2} d^{3} x^{2} - 6 \, a b d^{3} x + a^{2} d^{3}\right )} e^{2} + {\left (3 \, b^{2} d^{4} x - 2 \, a b d^{4}\right )} e\right )}} \end {gather*}

2/15*(3*B*b^2*d*x^2 + (B*a*b + 5*A*b^2)*d*x - (2*B*a^2 - 5*A*a*b)*d - (3*A*a^2 + (5*B*a*b - 2*A*b^2)*x^2 + (5*

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

^2*d^2*x^3 - 6*a*b*d^2*x^2 + 3*a^2*d^2*x)*e^3 + (3*b^2*d^3*x^2 - 6*a*b*d^3*x + a^2*d^3)*e^2 + (3*b^2*d^4*x - 2

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (88) = 176\).
time = 2.10, size = 180, normalized size = 1.89 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} {\left (\frac {{\left (3 \, B b^{6} d {\left | b \right |} e^{2} - 5 \, B a b^{5} {\left | b \right |} e^{3} + 2 \, A b^{6} {\left | b \right |} e^{3}\right )} {\left (b x + a\right )}}{b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}} - \frac {5 \, {\left (B a b^{6} d {\left | b \right |} e^{2} - A b^{7} d {\left | b \right |} e^{2} - B a^{2} b^{5} {\left | b \right |} e^{3} + A a b^{6} {\left | b \right |} e^{3}\right )}}{b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}}\right )}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \end {gather*}

2/15*(b*x + a)^(3/2)*((3*B*b^6*d*abs(b)*e^2 - 5*B*a*b^5*abs(b)*e^3 + 2*A*b^6*abs(b)*e^3)*(b*x + a)/(b^4*d^2*e^

2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4) - 5*(B*a*b^6*d*abs(b)*e^2 - A*b^7*d*abs(b)*e^2 - B*a^2*b^5*abs(b)*e^3 + A*a*b

^6*abs(b)*e^3)/(b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4))/(b^2*d + (b*x + a)*b*e - a*b*e)^(5/2)

________________________________________________________________________________________

Mupad [B]
time = 1.90, size = 179, normalized size = 1.88 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {x^2\,\sqrt {a+b\,x}\,\left (4\,A\,b^2\,e+6\,B\,b^2\,d-10\,B\,a\,b\,e\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^2}-\frac {\sqrt {a+b\,x}\,\left (6\,A\,a^2\,e+4\,B\,a^2\,d-10\,A\,a\,b\,d\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^2}+\frac {x\,\sqrt {a+b\,x}\,\left (10\,A\,b^2\,d-10\,B\,a^2\,e-2\,A\,a\,b\,e+2\,B\,a\,b\,d\right )}{15\,e^3\,{\left (a\,e-b\,d\right )}^2}\right )}{x^3+\frac {d^3}{e^3}+\frac {3\,d\,x^2}{e}+\frac {3\,d^2\,x}{e^2}} \end {gather*}

((d + e*x)^(1/2)*((x^2*(a + b*x)^(1/2)*(4*A*b^2*e + 6*B*b^2*d - 10*B*a*b*e))/(15*e^3*(a*e - b*d)^2) - ((a + b*

x)^(1/2)*(6*A*a^2*e + 4*B*a^2*d - 10*A*a*b*d))/(15*e^3*(a*e - b*d)^2) + (x*(a + b*x)^(1/2)*(10*A*b^2*d - 10*B*

a^2*e - 2*A*a*b*e + 2*B*a*b*d))/(15*e^3*(a*e - b*d)^2)))/(x^3 + d^3/e^3 + (3*d*x^2)/e + (3*d^2*x)/e^2)

________________________________________________________________________________________

3.23.6 \(\int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{7/2}} \, dx\) [2206]