\(\int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx\) [2210]

Optimal result

Integrand size = 24, antiderivative size = 309 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {128 b^3 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}+\frac {256 b^4 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{45045 e (b d-a e)^6 (d+e x)^{3/2}} \]

[Out]

-2/13*(-A*e+B*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)/(e*x+d)^(13/2)+2/143*(10*A*b*e-13*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(
-a*e+b*d)^2/(e*x+d)^(11/2)+16/1287*b*(10*A*b*e-13*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^3/(e*x+d)^(9/2)+32
/3003*b^2*(10*A*b*e-13*B*a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^4/(e*x+d)^(7/2)+128/15015*b^3*(10*A*b*e-13*B*
a*e+3*B*b*d)*(b*x+a)^(3/2)/e/(-a*e+b*d)^5/(e*x+d)^(5/2)+256/45045*b^4*(10*A*b*e-13*B*a*e+3*B*b*d)*(b*x+a)^(3/2
)/e/(-a*e+b*d)^6/(e*x+d)^(3/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {256 b^4 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{45045 e (d+e x)^{3/2} (b d-a e)^6}+\frac {128 b^3 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac {32 b^2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac {16 b (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-13 a B e+10 A b e+3 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

[In]

Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a +
b*x)^(3/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2)) + (16*b*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(
1287*e*(b*d - a*e)^3*(d + e*x)^(9/2)) + (32*b^2*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(3003*e*(b*d
- a*e)^4*(d + e*x)^(7/2)) + (128*b^3*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(15015*e*(b*d - a*e)^5*(
d + e*x)^(5/2)) + (256*b^4*(3*b*B*d + 10*A*b*e - 13*a*B*e)*(a + b*x)^(3/2))/(45045*e*(b*d - a*e)^6*(d + e*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 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {(3 b B d+10 A b e-13 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)} \\ & = -\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {(8 b (3 b B d+10 A b e-13 a B e)) \int \frac {\sqrt {a+b x}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2} \\ & = -\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {\left (16 b^2 (3 b B d+10 A b e-13 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{9/2}} \, dx}{429 e (b d-a e)^3} \\ & = -\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {\left (64 b^3 (3 b B d+10 A b e-13 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}} \, dx}{3003 e (b d-a e)^4} \\ & = -\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {128 b^3 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}+\frac {\left (128 b^4 (3 b B d+10 A b e-13 a B e)\right ) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{15015 e (b d-a e)^5} \\ & = -\frac {2 (B d-A e) (a+b x)^{3/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {16 b (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {32 b^2 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{3003 e (b d-a e)^4 (d+e x)^{7/2}}+\frac {128 b^3 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{15015 e (b d-a e)^5 (d+e x)^{5/2}}+\frac {256 b^4 (3 b B d+10 A b e-13 a B e) (a+b x)^{3/2}}{45045 e (b d-a e)^6 (d+e x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {2 (a+b x)^{3/2} \left (3465 B d e^4 (a+b x)^5-3465 A e^5 (a+b x)^5-16380 b B d e^3 (a+b x)^4 (d+e x)+20475 A b e^4 (a+b x)^4 (d+e x)-4095 a B e^4 (a+b x)^4 (d+e x)+30030 b^2 B d e^2 (a+b x)^3 (d+e x)^2-50050 A b^2 e^3 (a+b x)^3 (d+e x)^2+20020 a b B e^3 (a+b x)^3 (d+e x)^2-25740 b^3 B d e (a+b x)^2 (d+e x)^3+64350 A b^3 e^2 (a+b x)^2 (d+e x)^3-38610 a b^2 B e^2 (a+b x)^2 (d+e x)^3+9009 b^4 B d (a+b x) (d+e x)^4-45045 A b^4 e (a+b x) (d+e x)^4+36036 a b^3 B e (a+b x) (d+e x)^4+15015 A b^5 (d+e x)^5-15015 a b^4 B (d+e x)^5\right )}{45045 (b d-a e)^6 (d+e x)^{13/2}} \]

[In]

Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*(a + b*x)^(3/2)*(3465*B*d*e^4*(a + b*x)^5 - 3465*A*e^5*(a + b*x)^5 - 16380*b*B*d*e^3*(a + b*x)^4*(d + e*x)
+ 20475*A*b*e^4*(a + b*x)^4*(d + e*x) - 4095*a*B*e^4*(a + b*x)^4*(d + e*x) + 30030*b^2*B*d*e^2*(a + b*x)^3*(d
+ e*x)^2 - 50050*A*b^2*e^3*(a + b*x)^3*(d + e*x)^2 + 20020*a*b*B*e^3*(a + b*x)^3*(d + e*x)^2 - 25740*b^3*B*d*e
*(a + b*x)^2*(d + e*x)^3 + 64350*A*b^3*e^2*(a + b*x)^2*(d + e*x)^3 - 38610*a*b^2*B*e^2*(a + b*x)^2*(d + e*x)^3
+ 9009*b^4*B*d*(a + b*x)*(d + e*x)^4 - 45045*A*b^4*e*(a + b*x)*(d + e*x)^4 + 36036*a*b^3*B*e*(a + b*x)*(d + e
*x)^4 + 15015*A*b^5*(d + e*x)^5 - 15015*a*b^4*B*(d + e*x)^5))/(45045*(b*d - a*e)^6*(d + e*x)^(13/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(652\) vs. \(2(273)=546\).

Time = 1.09 (sec) , antiderivative size = 653, normalized size of antiderivative = 2.11

method	result	size
default	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} e^{5} x^{5}+1664 B a \,b^{4} e^{5} x^{5}-384 B \,b^{5} d \,e^{4} x^{5}+1920 A a \,b^{4} e^{5} x^{4}-8320 A \,b^{5} d \,e^{4} x^{4}-2496 B \,a^{2} b^{3} e^{5} x^{4}+11392 B a \,b^{4} d \,e^{4} x^{4}-2496 B \,b^{5} d^{2} e^{3} x^{4}-2400 A \,a^{2} b^{3} e^{5} x^{3}+12480 A a \,b^{4} d \,e^{4} x^{3}-22880 A \,b^{5} d^{2} e^{3} x^{3}+3120 B \,a^{3} b^{2} e^{5} x^{3}-16944 B \,a^{2} b^{3} d \,e^{4} x^{3}+33488 B a \,b^{4} d^{2} e^{3} x^{3}-6864 B \,b^{5} d^{3} e^{2} x^{3}+2800 A \,a^{3} b^{2} e^{5} x^{2}-15600 A \,a^{2} b^{3} d \,e^{4} x^{2}+34320 A a \,b^{4} d^{2} e^{3} x^{2}-34320 A \,b^{5} d^{3} e^{2} x^{2}-3640 B \,a^{4} b \,e^{5} x^{2}+21120 B \,a^{3} b^{2} d \,e^{4} x^{2}-49296 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+54912 B a \,b^{4} d^{3} e^{2} x^{2}-10296 B \,b^{5} d^{4} e \,x^{2}-3150 A \,a^{4} b \,e^{5} x +18200 A \,a^{3} b^{2} d \,e^{4} x -42900 A \,a^{2} b^{3} d^{2} e^{3} x +51480 A a \,b^{4} d^{3} e^{2} x -30030 A \,b^{5} d^{4} e x +4095 B \,a^{5} e^{5} x -24605 B \,a^{4} b d \,e^{4} x +61230 B \,a^{3} b^{2} d^{2} e^{3} x -79794 B \,a^{2} b^{3} d^{3} e^{2} x +54483 B a \,b^{4} d^{4} e x -9009 B \,b^{5} d^{5} x +3465 A \,a^{5} e^{5}-20475 A \,a^{4} b d \,e^{4}+50050 A \,a^{3} b^{2} d^{2} e^{3}-64350 A \,a^{2} b^{3} d^{3} e^{2}+45045 A a \,b^{4} d^{4} e -15015 A \,b^{5} d^{5}+630 B \,a^{5} d \,e^{4}-3640 B \,a^{4} b \,d^{2} e^{3}+8580 B \,a^{3} b^{2} d^{3} e^{2}-10296 B \,a^{2} b^{3} d^{4} e +6006 B a \,b^{4} d^{5}\right )}{45045 \left (e x +d \right )^{\frac {13}{2}} \left (a e -b d \right )^{6}}\)	\(653\)
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} e^{5} x^{5}+1664 B a \,b^{4} e^{5} x^{5}-384 B \,b^{5} d \,e^{4} x^{5}+1920 A a \,b^{4} e^{5} x^{4}-8320 A \,b^{5} d \,e^{4} x^{4}-2496 B \,a^{2} b^{3} e^{5} x^{4}+11392 B a \,b^{4} d \,e^{4} x^{4}-2496 B \,b^{5} d^{2} e^{3} x^{4}-2400 A \,a^{2} b^{3} e^{5} x^{3}+12480 A a \,b^{4} d \,e^{4} x^{3}-22880 A \,b^{5} d^{2} e^{3} x^{3}+3120 B \,a^{3} b^{2} e^{5} x^{3}-16944 B \,a^{2} b^{3} d \,e^{4} x^{3}+33488 B a \,b^{4} d^{2} e^{3} x^{3}-6864 B \,b^{5} d^{3} e^{2} x^{3}+2800 A \,a^{3} b^{2} e^{5} x^{2}-15600 A \,a^{2} b^{3} d \,e^{4} x^{2}+34320 A a \,b^{4} d^{2} e^{3} x^{2}-34320 A \,b^{5} d^{3} e^{2} x^{2}-3640 B \,a^{4} b \,e^{5} x^{2}+21120 B \,a^{3} b^{2} d \,e^{4} x^{2}-49296 B \,a^{2} b^{3} d^{2} e^{3} x^{2}+54912 B a \,b^{4} d^{3} e^{2} x^{2}-10296 B \,b^{5} d^{4} e \,x^{2}-3150 A \,a^{4} b \,e^{5} x +18200 A \,a^{3} b^{2} d \,e^{4} x -42900 A \,a^{2} b^{3} d^{2} e^{3} x +51480 A a \,b^{4} d^{3} e^{2} x -30030 A \,b^{5} d^{4} e x +4095 B \,a^{5} e^{5} x -24605 B \,a^{4} b d \,e^{4} x +61230 B \,a^{3} b^{2} d^{2} e^{3} x -79794 B \,a^{2} b^{3} d^{3} e^{2} x +54483 B a \,b^{4} d^{4} e x -9009 B \,b^{5} d^{5} x +3465 A \,a^{5} e^{5}-20475 A \,a^{4} b d \,e^{4}+50050 A \,a^{3} b^{2} d^{2} e^{3}-64350 A \,a^{2} b^{3} d^{3} e^{2}+45045 A a \,b^{4} d^{4} e -15015 A \,b^{5} d^{5}+630 B \,a^{5} d \,e^{4}-3640 B \,a^{4} b \,d^{2} e^{3}+8580 B \,a^{3} b^{2} d^{3} e^{2}-10296 B \,a^{2} b^{3} d^{4} e +6006 B a \,b^{4} d^{5}\right )}{45045 \left (e x +d \right )^{\frac {13}{2}} \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}\)	\(722\)

[In]

int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(15/2),x,method=_RETURNVERBOSE)

[Out]

-2/45045*(b*x+a)^(3/2)/(e*x+d)^(13/2)*(-1280*A*b^5*e^5*x^5+1664*B*a*b^4*e^5*x^5-384*B*b^5*d*e^4*x^5+1920*A*a*b
^4*e^5*x^4-8320*A*b^5*d*e^4*x^4-2496*B*a^2*b^3*e^5*x^4+11392*B*a*b^4*d*e^4*x^4-2496*B*b^5*d^2*e^3*x^4-2400*A*a
^2*b^3*e^5*x^3+12480*A*a*b^4*d*e^4*x^3-22880*A*b^5*d^2*e^3*x^3+3120*B*a^3*b^2*e^5*x^3-16944*B*a^2*b^3*d*e^4*x^
3+33488*B*a*b^4*d^2*e^3*x^3-6864*B*b^5*d^3*e^2*x^3+2800*A*a^3*b^2*e^5*x^2-15600*A*a^2*b^3*d*e^4*x^2+34320*A*a*
b^4*d^2*e^3*x^2-34320*A*b^5*d^3*e^2*x^2-3640*B*a^4*b*e^5*x^2+21120*B*a^3*b^2*d*e^4*x^2-49296*B*a^2*b^3*d^2*e^3
*x^2+54912*B*a*b^4*d^3*e^2*x^2-10296*B*b^5*d^4*e*x^2-3150*A*a^4*b*e^5*x+18200*A*a^3*b^2*d*e^4*x-42900*A*a^2*b^
3*d^2*e^3*x+51480*A*a*b^4*d^3*e^2*x-30030*A*b^5*d^4*e*x+4095*B*a^5*e^5*x-24605*B*a^4*b*d*e^4*x+61230*B*a^3*b^2
*d^2*e^3*x-79794*B*a^2*b^3*d^3*e^2*x+54483*B*a*b^4*d^4*e*x-9009*B*b^5*d^5*x+3465*A*a^5*e^5-20475*A*a^4*b*d*e^4
+50050*A*a^3*b^2*d^2*e^3-64350*A*a^2*b^3*d^3*e^2+45045*A*a*b^4*d^4*e-15015*A*b^5*d^5+630*B*a^5*d*e^4-3640*B*a^
4*b*d^2*e^3+8580*B*a^3*b^2*d^3*e^2-10296*B*a^2*b^3*d^4*e+6006*B*a*b^4*d^5)/(a*e-b*d)^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (273) = 546\).

Time = 123.81 (sec) , antiderivative size = 1427, normalized size of antiderivative = 4.62 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(15/2),x, algorithm="fricas")

[Out]

-2/45045*(3465*A*a^6*e^5 - 128*(3*B*b^6*d*e^4 - (13*B*a*b^5 - 10*A*b^6)*e^5)*x^6 + 3003*(2*B*a^2*b^4 - 5*A*a*b
^5)*d^5 - 1287*(8*B*a^3*b^3 - 35*A*a^2*b^4)*d^4*e + 4290*(2*B*a^4*b^2 - 15*A*a^3*b^3)*d^3*e^2 - 910*(4*B*a^5*b
- 55*A*a^4*b^2)*d^2*e^3 + 315*(2*B*a^6 - 65*A*a^5*b)*d*e^4 - 64*(39*B*b^6*d^2*e^3 - 2*(86*B*a*b^5 - 65*A*b^6)
*d*e^4 + (13*B*a^2*b^4 - 10*A*a*b^5)*e^5)*x^5 - 16*(429*B*b^6*d^3*e^2 - 13*(149*B*a*b^5 - 110*A*b^6)*d^2*e^3 +
(347*B*a^2*b^4 - 260*A*a*b^5)*d*e^4 - 3*(13*B*a^3*b^3 - 10*A*a^2*b^4)*e^5)*x^4 - 8*(1287*B*b^6*d^4*e - 858*(7
*B*a*b^5 - 5*A*b^6)*d^3*e^2 + 26*(76*B*a^2*b^4 - 55*A*a*b^5)*d^2*e^3 - 6*(87*B*a^3*b^3 - 65*A*a^2*b^4)*d*e^4 +
5*(13*B*a^4*b^2 - 10*A*a^3*b^3)*e^5)*x^3 - (9009*B*b^6*d^5 - 429*(103*B*a*b^5 - 70*A*b^6)*d^4*e + 858*(29*B*a
^2*b^4 - 20*A*a*b^5)*d^3*e^2 - 78*(153*B*a^3*b^3 - 110*A*a^2*b^4)*d^2*e^3 + 5*(697*B*a^4*b^2 - 520*A*a^3*b^3)*
d*e^4 - 35*(13*B*a^5*b - 10*A*a^4*b^2)*e^5)*x^2 - (3003*(B*a*b^5 + 5*A*b^6)*d^5 - 429*(103*B*a^2*b^4 + 35*A*a*
b^5)*d^4*e + 858*(83*B*a^3*b^3 + 15*A*a^2*b^4)*d^3*e^2 - 130*(443*B*a^4*b^2 + 55*A*a^3*b^3)*d^2*e^3 + 175*(137
*B*a^5*b + 13*A*a^4*b^2)*d*e^4 - 315*(13*B*a^6 + A*a^5*b)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^6*d^13 - 6*a*
b^5*d^12*e + 15*a^2*b^4*d^11*e^2 - 20*a^3*b^3*d^10*e^3 + 15*a^4*b^2*d^9*e^4 - 6*a^5*b*d^8*e^5 + a^6*d^7*e^6 +
(b^6*d^6*e^7 - 6*a*b^5*d^5*e^8 + 15*a^2*b^4*d^4*e^9 - 20*a^3*b^3*d^3*e^10 + 15*a^4*b^2*d^2*e^11 - 6*a^5*b*d*e^
12 + a^6*e^13)*x^7 + 7*(b^6*d^7*e^6 - 6*a*b^5*d^6*e^7 + 15*a^2*b^4*d^5*e^8 - 20*a^3*b^3*d^4*e^9 + 15*a^4*b^2*d
^3*e^10 - 6*a^5*b*d^2*e^11 + a^6*d*e^12)*x^6 + 21*(b^6*d^8*e^5 - 6*a*b^5*d^7*e^6 + 15*a^2*b^4*d^6*e^7 - 20*a^3
*b^3*d^5*e^8 + 15*a^4*b^2*d^4*e^9 - 6*a^5*b*d^3*e^10 + a^6*d^2*e^11)*x^5 + 35*(b^6*d^9*e^4 - 6*a*b^5*d^8*e^5 +
15*a^2*b^4*d^7*e^6 - 20*a^3*b^3*d^6*e^7 + 15*a^4*b^2*d^5*e^8 - 6*a^5*b*d^4*e^9 + a^6*d^3*e^10)*x^4 + 35*(b^6*
d^10*e^3 - 6*a*b^5*d^9*e^4 + 15*a^2*b^4*d^8*e^5 - 20*a^3*b^3*d^7*e^6 + 15*a^4*b^2*d^6*e^7 - 6*a^5*b*d^5*e^8 +
a^6*d^4*e^9)*x^3 + 21*(b^6*d^11*e^2 - 6*a*b^5*d^10*e^3 + 15*a^2*b^4*d^9*e^4 - 20*a^3*b^3*d^8*e^5 + 15*a^4*b^2*
d^7*e^6 - 6*a^5*b*d^6*e^7 + a^6*d^5*e^8)*x^2 + 7*(b^6*d^12*e - 6*a*b^5*d^11*e^2 + 15*a^2*b^4*d^10*e^3 - 20*a^3
*b^3*d^9*e^4 + 15*a^4*b^2*d^8*e^5 - 6*a^5*b*d^7*e^6 + a^6*d^6*e^7)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(15/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(15/2),x, algorithm="maxima")

[Out]

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(e*(a*e-b*d)>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1303 vs. \(2 (273) = 546\).

Time = 0.75 (sec) , antiderivative size = 1303, normalized size of antiderivative = 4.22 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(15/2),x, algorithm="giac")

[Out]

2/45045*((8*(2*(4*(b*x + a)*(2*(3*B*b^14*d*e^10*abs(b) - 13*B*a*b^13*e^11*abs(b) + 10*A*b^14*e^11*abs(b))*(b*x
+ a)/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b
^3*d*e^11 + a^6*b^2*e^12) + 13*(3*B*b^15*d^2*e^9*abs(b) - 16*B*a*b^14*d*e^10*abs(b) + 10*A*b^15*d*e^10*abs(b)
+ 13*B*a^2*b^13*e^11*abs(b) - 10*A*a*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 2
0*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12)) + 143*(3*B*b^16*d^3*e^8*abs(b) - 1
9*B*a*b^15*d^2*e^9*abs(b) + 10*A*b^16*d^2*e^9*abs(b) + 29*B*a^2*b^14*d*e^10*abs(b) - 20*A*a*b^15*d*e^10*abs(b)
- 13*B*a^3*b^13*e^11*abs(b) + 10*A*a^2*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8
- 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x + a) + 429*(3*B*b^17*d^4*e
^7*abs(b) - 22*B*a*b^16*d^3*e^8*abs(b) + 10*A*b^17*d^3*e^8*abs(b) + 48*B*a^2*b^15*d^2*e^9*abs(b) - 30*A*a*b^16
*d^2*e^9*abs(b) - 42*B*a^3*b^14*d*e^10*abs(b) + 30*A*a^2*b^15*d*e^10*abs(b) + 13*B*a^4*b^13*e^11*abs(b) - 10*A
*a^3*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d
^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x + a) + 3003*(3*B*b^18*d^5*e^6*abs(b) - 25*B*a*b^17*d^4*e^7*ab
s(b) + 10*A*b^18*d^4*e^7*abs(b) + 70*B*a^2*b^16*d^3*e^8*abs(b) - 40*A*a*b^17*d^3*e^8*abs(b) - 90*B*a^3*b^15*d^
2*e^9*abs(b) + 60*A*a^2*b^16*d^2*e^9*abs(b) + 55*B*a^4*b^14*d*e^10*abs(b) - 40*A*a^3*b^15*d*e^10*abs(b) - 13*B
*a^5*b^13*e^11*abs(b) + 10*A*a^4*b^14*e^11*abs(b))/(b^8*d^6*e^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^
3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*b^2*e^12))*(b*x + a) - 15015*(B*a*b^18*d^5*e^6*ab
s(b) - A*b^19*d^5*e^6*abs(b) - 5*B*a^2*b^17*d^4*e^7*abs(b) + 5*A*a*b^18*d^4*e^7*abs(b) + 10*B*a^3*b^16*d^3*e^8
*abs(b) - 10*A*a^2*b^17*d^3*e^8*abs(b) - 10*B*a^4*b^15*d^2*e^9*abs(b) + 10*A*a^3*b^16*d^2*e^9*abs(b) + 5*B*a^5
*b^14*d*e^10*abs(b) - 5*A*a^4*b^15*d*e^10*abs(b) - B*a^6*b^13*e^11*abs(b) + A*a^5*b^14*e^11*abs(b))/(b^8*d^6*e
^6 - 6*a*b^7*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 6*a^5*b^3*d*e^11 + a^6*
b^2*e^12))*(b*x + a)^(3/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(13/2)

Mupad [B] (verification not implemented)

Time = 3.36 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.43 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {x\,\sqrt {a+b\,x}\,\left (-8190\,B\,a^6\,e^5+47950\,B\,a^5\,b\,d\,e^4-630\,A\,a^5\,b\,e^5-115180\,B\,a^4\,b^2\,d^2\,e^3+4550\,A\,a^4\,b^2\,d\,e^4+142428\,B\,a^3\,b^3\,d^3\,e^2-14300\,A\,a^3\,b^3\,d^2\,e^3-88374\,B\,a^2\,b^4\,d^4\,e+25740\,A\,a^2\,b^4\,d^3\,e^2+6006\,B\,a\,b^5\,d^5-30030\,A\,a\,b^5\,d^4\,e+30030\,A\,b^6\,d^5\right )}{45045\,e^7\,{\left (a\,e-b\,d\right )}^6}-\frac {\sqrt {a+b\,x}\,\left (1260\,B\,a^6\,d\,e^4+6930\,A\,a^6\,e^5-7280\,B\,a^5\,b\,d^2\,e^3-40950\,A\,a^5\,b\,d\,e^4+17160\,B\,a^4\,b^2\,d^3\,e^2+100100\,A\,a^4\,b^2\,d^2\,e^3-20592\,B\,a^3\,b^3\,d^4\,e-128700\,A\,a^3\,b^3\,d^3\,e^2+12012\,B\,a^2\,b^4\,d^5+90090\,A\,a^2\,b^4\,d^4\,e-30030\,A\,a\,b^5\,d^5\right )}{45045\,e^7\,{\left (a\,e-b\,d\right )}^6}+\frac {256\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )}{45045\,e^3\,{\left (a\,e-b\,d\right )}^6}-\frac {16\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )\,\left (5\,a^3\,e^3-39\,a^2\,b\,d\,e^2+143\,a\,b^2\,d^2\,e-429\,b^3\,d^3\right )}{45045\,e^6\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,x^2\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )\,\left (35\,a^4\,e^4-260\,a^3\,b\,d\,e^3+858\,a^2\,b^2\,d^2\,e^2-1716\,a\,b^3\,d^3\,e+3003\,b^4\,d^4\right )}{45045\,e^7\,{\left (a\,e-b\,d\right )}^6}-\frac {128\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )}{45045\,e^4\,{\left (a\,e-b\,d\right )}^6}+\frac {32\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (10\,A\,b\,e-13\,B\,a\,e+3\,B\,b\,d\right )}{45045\,e^5\,{\left (a\,e-b\,d\right )}^6}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \]

[In]

int(((A + B*x)*(a + b*x)^(1/2))/(d + e*x)^(15/2),x)

[Out]

((d + e*x)^(1/2)*((x*(a + b*x)^(1/2)*(30030*A*b^6*d^5 - 8190*B*a^6*e^5 - 630*A*a^5*b*e^5 + 6006*B*a*b^5*d^5 +
4550*A*a^4*b^2*d*e^4 - 88374*B*a^2*b^4*d^4*e + 25740*A*a^2*b^4*d^3*e^2 - 14300*A*a^3*b^3*d^2*e^3 + 142428*B*a^
3*b^3*d^3*e^2 - 115180*B*a^4*b^2*d^2*e^3 - 30030*A*a*b^5*d^4*e + 47950*B*a^5*b*d*e^4))/(45045*e^7*(a*e - b*d)^
6) - ((a + b*x)^(1/2)*(6930*A*a^6*e^5 - 30030*A*a*b^5*d^5 + 1260*B*a^6*d*e^4 + 12012*B*a^2*b^4*d^5 + 90090*A*a
^2*b^4*d^4*e - 20592*B*a^3*b^3*d^4*e - 7280*B*a^5*b*d^2*e^3 - 128700*A*a^3*b^3*d^3*e^2 + 100100*A*a^4*b^2*d^2*
e^3 + 17160*B*a^4*b^2*d^3*e^2 - 40950*A*a^5*b*d*e^4))/(45045*e^7*(a*e - b*d)^6) + (256*b^5*x^6*(a + b*x)^(1/2)
*(10*A*b*e - 13*B*a*e + 3*B*b*d))/(45045*e^3*(a*e - b*d)^6) - (16*b^2*x^3*(a + b*x)^(1/2)*(10*A*b*e - 13*B*a*e
+ 3*B*b*d)*(5*a^3*e^3 - 429*b^3*d^3 + 143*a*b^2*d^2*e - 39*a^2*b*d*e^2))/(45045*e^6*(a*e - b*d)^6) + (2*b*x^2
*(a + b*x)^(1/2)*(10*A*b*e - 13*B*a*e + 3*B*b*d)*(35*a^4*e^4 + 3003*b^4*d^4 + 858*a^2*b^2*d^2*e^2 - 1716*a*b^3
*d^3*e - 260*a^3*b*d*e^3))/(45045*e^7*(a*e - b*d)^6) - (128*b^4*x^5*(a*e - 13*b*d)*(a + b*x)^(1/2)*(10*A*b*e -
13*B*a*e + 3*B*b*d))/(45045*e^4*(a*e - b*d)^6) + (32*b^3*x^4*(a + b*x)^(1/2)*(3*a^2*e^2 + 143*b^2*d^2 - 26*a*
b*d*e)*(10*A*b*e - 13*B*a*e + 3*B*b*d))/(45045*e^5*(a*e - b*d)^6)))/(x^7 + d^7/e^7 + (7*d*x^6)/e + (7*d^6*x)/e
^6 + (21*d^2*x^5)/e^2 + (35*d^3*x^4)/e^3 + (35*d^4*x^3)/e^4 + (21*d^5*x^2)/e^5)