\(\int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx\) [2234]

Optimal result

Integrand size = 24, antiderivative size = 255 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {16 b^2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{6435 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {32 b^3 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{45045 e (b d-a e)^5 (d+e x)^{7/2}} \]

[Out]

-2/15*(-A*e+B*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)/(e*x+d)^(15/2)+2/195*(8*A*b*e-15*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-
a*e+b*d)^2/(e*x+d)^(13/2)+4/715*b*(8*A*b*e-15*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^3/(e*x+d)^(11/2)+16/64
35*b^2*(8*A*b*e-15*B*a*e+7*B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^4/(e*x+d)^(9/2)+32/45045*b^3*(8*A*b*e-15*B*a*e+7*
B*b*d)*(b*x+a)^(7/2)/e/(-a*e+b*d)^5/(e*x+d)^(7/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\frac {32 b^3 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{45045 e (d+e x)^{7/2} (b d-a e)^5}+\frac {16 b^2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{6435 e (d+e x)^{9/2} (b d-a e)^4}+\frac {4 b (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{715 e (d+e x)^{11/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{195 e (d+e x)^{13/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a +
b*x)^(7/2))/(195*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (4*b*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(715
*e*(b*d - a*e)^3*(d + e*x)^(11/2)) + (16*b^2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(6435*e*(b*d - a*
e)^4*(d + e*x)^(9/2)) + (32*b^3*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(45045*e*(b*d - a*e)^5*(d + e*
x)^(7/2))

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)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {(7 b B d+8 A b e-15 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{15/2}} \, dx}{15 e (b d-a e)} \\ & = -\frac {2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {(2 b (7 b B d+8 A b e-15 a B e)) \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{65 e (b d-a e)^2} \\ & = -\frac {2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {\left (8 b^2 (7 b B d+8 A b e-15 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{715 e (b d-a e)^3} \\ & = -\frac {2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {16 b^2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{6435 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {\left (16 b^3 (7 b B d+8 A b e-15 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{6435 e (b d-a e)^4} \\ & = -\frac {2 (B d-A e) (a+b x)^{7/2}}{15 e (b d-a e) (d+e x)^{15/2}}+\frac {2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{195 e (b d-a e)^2 (d+e x)^{13/2}}+\frac {4 b (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{715 e (b d-a e)^3 (d+e x)^{11/2}}+\frac {16 b^2 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{6435 e (b d-a e)^4 (d+e x)^{9/2}}+\frac {32 b^3 (7 b B d+8 A b e-15 a B e) (a+b x)^{7/2}}{45045 e (b d-a e)^5 (d+e x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\frac {2 (a+b x)^{7/2} \left (-3003 B d e^3 (a+b x)^4+3003 A e^4 (a+b x)^4+10395 b B d e^2 (a+b x)^3 (d+e x)-13860 A b e^3 (a+b x)^3 (d+e x)+3465 a B e^3 (a+b x)^3 (d+e x)-12285 b^2 B d e (a+b x)^2 (d+e x)^2+24570 A b^2 e^2 (a+b x)^2 (d+e x)^2-12285 a b B e^2 (a+b x)^2 (d+e x)^2+5005 b^3 B d (a+b x) (d+e x)^3-20020 A b^3 e (a+b x) (d+e x)^3+15015 a b^2 B e (a+b x) (d+e x)^3+6435 A b^4 (d+e x)^4-6435 a b^3 B (d+e x)^4\right )}{45045 (b d-a e)^5 (d+e x)^{15/2}} \]

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(2*(a + b*x)^(7/2)*(-3003*B*d*e^3*(a + b*x)^4 + 3003*A*e^4*(a + b*x)^4 + 10395*b*B*d*e^2*(a + b*x)^3*(d + e*x)
- 13860*A*b*e^3*(a + b*x)^3*(d + e*x) + 3465*a*B*e^3*(a + b*x)^3*(d + e*x) - 12285*b^2*B*d*e*(a + b*x)^2*(d +
e*x)^2 + 24570*A*b^2*e^2*(a + b*x)^2*(d + e*x)^2 - 12285*a*b*B*e^2*(a + b*x)^2*(d + e*x)^2 + 5005*b^3*B*d*(a
+ b*x)*(d + e*x)^3 - 20020*A*b^3*e*(a + b*x)*(d + e*x)^3 + 15015*a*b^2*B*e*(a + b*x)*(d + e*x)^3 + 6435*A*b^4*
(d + e*x)^4 - 6435*a*b^3*B*(d + e*x)^4))/(45045*(b*d - a*e)^5*(d + e*x)^(15/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(225)=450\).

Time = 1.09 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.98

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (128 A \,b^{4} e^{4} x^{4}-240 B a \,b^{3} e^{4} x^{4}+112 B \,b^{4} d \,e^{3} x^{4}-448 A a \,b^{3} e^{4} x^{3}+960 A \,b^{4} d \,e^{3} x^{3}+840 B \,a^{2} b^{2} e^{4} x^{3}-2192 B a \,b^{3} d \,e^{3} x^{3}+840 B \,b^{4} d^{2} e^{2} x^{3}+1008 A \,a^{2} b^{2} e^{4} x^{2}-3360 A a \,b^{3} d \,e^{3} x^{2}+3120 A \,b^{4} d^{2} e^{2} x^{2}-1890 B \,a^{3} b \,e^{4} x^{2}+7182 B \,a^{2} b^{2} d \,e^{3} x^{2}-8790 B a \,b^{3} d^{2} e^{2} x^{2}+2730 B \,b^{4} d^{3} e \,x^{2}-1848 A \,a^{3} b \,e^{4} x +7560 A \,a^{2} b^{2} d \,e^{3} x -10920 A a \,b^{3} d^{2} e^{2} x +5720 A \,b^{4} d^{3} e x +3465 B \,a^{4} e^{4} x -15792 B \,a^{3} b d \,e^{3} x +27090 B \,a^{2} b^{2} d^{2} e^{2} x -20280 B a \,b^{3} d^{3} e x +5005 B \,b^{4} d^{4} x +3003 A \,a^{4} e^{4}-13860 A \,a^{3} b d \,e^{3}+24570 A \,a^{2} b^{2} d^{2} e^{2}-20020 A a \,b^{3} d^{3} e +6435 A \,b^{4} d^{4}+462 B \,a^{4} d \,e^{3}-1890 B \,a^{3} b \,d^{2} e^{2}+2730 B \,a^{2} b^{2} d^{3} e -1430 B a \,b^{3} d^{4}\right )}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}\)	\(505\)
default	\(-\frac {2 \left (128 A \,b^{6} e^{4} x^{6}-240 B a \,b^{5} e^{4} x^{6}+112 B \,b^{6} d \,e^{3} x^{6}-192 A a \,b^{5} e^{4} x^{5}+960 A \,b^{6} d \,e^{3} x^{5}+360 B \,a^{2} b^{4} e^{4} x^{5}-1968 B a \,b^{5} d \,e^{3} x^{5}+840 B \,b^{6} d^{2} e^{2} x^{5}+240 A \,a^{2} b^{4} e^{4} x^{4}-1440 A a \,b^{5} d \,e^{3} x^{4}+3120 A \,b^{6} d^{2} e^{2} x^{4}-450 B \,a^{3} b^{3} e^{4} x^{4}+2910 B \,a^{2} b^{4} d \,e^{3} x^{4}-7110 B a \,b^{5} d^{2} e^{2} x^{4}+2730 B \,b^{6} d^{3} e \,x^{4}-280 A \,a^{3} b^{3} e^{4} x^{3}+1800 A \,a^{2} b^{4} d \,e^{3} x^{3}-4680 A a \,b^{5} d^{2} e^{2} x^{3}+5720 A \,b^{6} d^{3} e \,x^{3}+525 B \,a^{4} b^{2} e^{4} x^{3}-3620 B \,a^{3} b^{3} d \,e^{3} x^{3}+10350 B \,a^{2} b^{4} d^{2} e^{2} x^{3}-14820 B a \,b^{5} d^{3} e \,x^{3}+5005 B \,b^{6} d^{4} x^{3}+315 A \,a^{4} b^{2} e^{4} x^{2}-2100 A \,a^{3} b^{3} d \,e^{3} x^{2}+5850 A \,a^{2} b^{4} d^{2} e^{2} x^{2}-8580 A a \,b^{5} d^{3} e \,x^{2}+6435 A \,b^{6} d^{4} x^{2}+5040 B \,a^{5} b \,e^{4} x^{2}-23940 B \,a^{4} b^{2} d \,e^{3} x^{2}+43500 B \,a^{3} b^{3} d^{2} e^{2} x^{2}-35100 B \,a^{2} b^{4} d^{3} e \,x^{2}+8580 B a \,b^{5} d^{4} x^{2}+4158 A \,a^{5} b \,e^{4} x -20160 A \,a^{4} b^{2} d \,e^{3} x +38220 A \,a^{3} b^{3} d^{2} e^{2} x -34320 A \,a^{2} b^{4} d^{3} e x +12870 A a \,b^{5} d^{4} x +3465 B \,a^{6} e^{4} x -14868 B \,a^{5} b d \,e^{3} x +23310 B \,a^{4} b^{2} d^{2} e^{2} x -14820 B \,a^{3} b^{3} d^{3} e x +2145 B \,a^{2} b^{4} d^{4} x +3003 A \,a^{6} e^{4}-13860 A \,a^{5} b d \,e^{3}+24570 A \,a^{4} b^{2} d^{2} e^{2}-20020 A \,a^{3} b^{3} d^{3} e +6435 A \,a^{2} b^{4} d^{4}+462 B \,a^{6} d \,e^{3}-1890 B \,a^{5} b \,d^{2} e^{2}+2730 B \,a^{4} b^{2} d^{3} e -1430 B \,a^{3} b^{3} d^{4}\right ) \left (b x +a \right )^{\frac {3}{2}}}{45045 \left (e x +d \right )^{\frac {15}{2}} \left (a e -b d \right )^{5}}\)	\(765\)

[In]

int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x,method=_RETURNVERBOSE)

[Out]

-2/45045*(b*x+a)^(7/2)*(128*A*b^4*e^4*x^4-240*B*a*b^3*e^4*x^4+112*B*b^4*d*e^3*x^4-448*A*a*b^3*e^4*x^3+960*A*b^
4*d*e^3*x^3+840*B*a^2*b^2*e^4*x^3-2192*B*a*b^3*d*e^3*x^3+840*B*b^4*d^2*e^2*x^3+1008*A*a^2*b^2*e^4*x^2-3360*A*a
*b^3*d*e^3*x^2+3120*A*b^4*d^2*e^2*x^2-1890*B*a^3*b*e^4*x^2+7182*B*a^2*b^2*d*e^3*x^2-8790*B*a*b^3*d^2*e^2*x^2+2
730*B*b^4*d^3*e*x^2-1848*A*a^3*b*e^4*x+7560*A*a^2*b^2*d*e^3*x-10920*A*a*b^3*d^2*e^2*x+5720*A*b^4*d^3*e*x+3465*
B*a^4*e^4*x-15792*B*a^3*b*d*e^3*x+27090*B*a^2*b^2*d^2*e^2*x-20280*B*a*b^3*d^3*e*x+5005*B*b^4*d^4*x+3003*A*a^4*
e^4-13860*A*a^3*b*d*e^3+24570*A*a^2*b^2*d^2*e^2-20020*A*a*b^3*d^3*e+6435*A*b^4*d^4+462*B*a^4*d*e^3-1890*B*a^3*
b*d^2*e^2+2730*B*a^2*b^2*d^3*e-1430*B*a*b^3*d^4)/(e*x+d)^(15/2)/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a
^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1465 vs. \(2 (225) = 450\).

Time = 248.01 (sec) , antiderivative size = 1465, normalized size of antiderivative = 5.75 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="fricas")

[Out]

2/45045*(3003*A*a^7*e^4 + 16*(7*B*b^7*d*e^3 - (15*B*a*b^6 - 8*A*b^7)*e^4)*x^7 + 8*(105*B*b^7*d^2*e^2 - 8*(29*B
*a*b^6 - 15*A*b^7)*d*e^3 + (15*B*a^2*b^5 - 8*A*a*b^6)*e^4)*x^6 + 6*(455*B*b^7*d^3*e - 5*(209*B*a*b^6 - 104*A*b
^7)*d^2*e^2 + (157*B*a^2*b^5 - 80*A*a*b^6)*d*e^3 - (15*B*a^3*b^4 - 8*A*a^2*b^5)*e^4)*x^5 - 715*(2*B*a^4*b^3 -
9*A*a^3*b^4)*d^4 + 910*(3*B*a^5*b^2 - 22*A*a^4*b^3)*d^3*e - 1890*(B*a^6*b - 13*A*a^5*b^2)*d^2*e^2 + 462*(B*a^7
- 30*A*a^6*b)*d*e^3 + 5*(1001*B*b^7*d^4 - 26*(93*B*a*b^6 - 44*A*b^7)*d^3*e + 24*(27*B*a^2*b^5 - 13*A*a*b^6)*d
^2*e^2 - 2*(71*B*a^3*b^4 - 36*A*a^2*b^5)*d*e^3 + (15*B*a^4*b^3 - 8*A*a^3*b^4)*e^4)*x^4 + 5*(143*(19*B*a*b^6 +
9*A*b^7)*d^4 - 52*(192*B*a^2*b^5 + 11*A*a*b^6)*d^3*e + 6*(1795*B*a^3*b^4 + 39*A*a^2*b^5)*d^2*e^2 - 4*(1378*B*a
^4*b^3 + 15*A*a^3*b^4)*d*e^3 + 7*(159*B*a^5*b^2 + A*a^4*b^3)*e^4)*x^3 + 3*(715*(5*B*a^2*b^5 + 9*A*a*b^6)*d^4 -
260*(64*B*a^3*b^4 + 55*A*a^2*b^5)*d^3*e + 10*(2227*B*a^4*b^3 + 1469*A*a^3*b^4)*d^2*e^2 - 28*(462*B*a^5*b^2 +
265*A*a^4*b^3)*d*e^3 + 21*(135*B*a^6*b + 71*A*a^5*b^2)*e^4)*x^2 + (715*(B*a^3*b^4 + 27*A*a^2*b^5)*d^4 - 130*(9
3*B*a^4*b^3 + 418*A*a^3*b^4)*d^3*e + 210*(102*B*a^5*b^2 + 299*A*a^4*b^3)*d^2*e^2 - 42*(343*B*a^6*b + 810*A*a^5
*b^2)*d*e^3 + 231*(15*B*a^7 + 31*A*a^6*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^5*d^13 - 5*a*b^4*d^12*e + 10*
a^2*b^3*d^11*e^2 - 10*a^3*b^2*d^10*e^3 + 5*a^4*b*d^9*e^4 - a^5*d^8*e^5 + (b^5*d^5*e^8 - 5*a*b^4*d^4*e^9 + 10*a
^2*b^3*d^3*e^10 - 10*a^3*b^2*d^2*e^11 + 5*a^4*b*d*e^12 - a^5*e^13)*x^8 + 8*(b^5*d^6*e^7 - 5*a*b^4*d^5*e^8 + 10
*a^2*b^3*d^4*e^9 - 10*a^3*b^2*d^3*e^10 + 5*a^4*b*d^2*e^11 - a^5*d*e^12)*x^7 + 28*(b^5*d^7*e^6 - 5*a*b^4*d^6*e^
7 + 10*a^2*b^3*d^5*e^8 - 10*a^3*b^2*d^4*e^9 + 5*a^4*b*d^3*e^10 - a^5*d^2*e^11)*x^6 + 56*(b^5*d^8*e^5 - 5*a*b^4
*d^7*e^6 + 10*a^2*b^3*d^6*e^7 - 10*a^3*b^2*d^5*e^8 + 5*a^4*b*d^4*e^9 - a^5*d^3*e^10)*x^5 + 70*(b^5*d^9*e^4 - 5
*a*b^4*d^8*e^5 + 10*a^2*b^3*d^7*e^6 - 10*a^3*b^2*d^6*e^7 + 5*a^4*b*d^5*e^8 - a^5*d^4*e^9)*x^4 + 56*(b^5*d^10*e
^3 - 5*a*b^4*d^9*e^4 + 10*a^2*b^3*d^8*e^5 - 10*a^3*b^2*d^7*e^6 + 5*a^4*b*d^6*e^7 - a^5*d^5*e^8)*x^3 + 28*(b^5*
d^11*e^2 - 5*a*b^4*d^10*e^3 + 10*a^2*b^3*d^9*e^4 - 10*a^3*b^2*d^8*e^5 + 5*a^4*b*d^7*e^6 - a^5*d^6*e^7)*x^2 + 8
*(b^5*d^12*e - 5*a*b^4*d^11*e^2 + 10*a^2*b^3*d^10*e^3 - 10*a^3*b^2*d^9*e^4 + 5*a^4*b*d^8*e^5 - a^5*d^7*e^6)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(17/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),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. 1413 vs. \(2 (225) = 450\).

Time = 1.26 (sec) , antiderivative size = 1413, normalized size of antiderivative = 5.54 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x, algorithm="giac")

[Out]

2/45045*((2*(4*(b*x + a)*(2*(7*B*b^18*d^3*e^10*abs(b) - 29*B*a*b^17*d^2*e^11*abs(b) + 8*A*b^18*d^2*e^11*abs(b)
+ 37*B*a^2*b^16*d*e^12*abs(b) - 16*A*a*b^17*d*e^12*abs(b) - 15*B*a^3*b^15*e^13*abs(b) + 8*A*a^2*b^16*e^13*abs
(b))*(b*x + a)/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11
- 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14) + 15*(7*B*b^19*d^4*e^9*abs(b) - 36*B*a*b^18*d^3*e^10
*abs(b) + 8*A*b^19*d^3*e^10*abs(b) + 66*B*a^2*b^17*d^2*e^11*abs(b) - 24*A*a*b^18*d^2*e^11*abs(b) - 52*B*a^3*b^
16*d*e^12*abs(b) + 24*A*a^2*b^17*d*e^12*abs(b) + 15*B*a^4*b^15*e^13*abs(b) - 8*A*a^3*b^16*e^13*abs(b))/(b^9*d^
7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12
+ 7*a^6*b^3*d*e^13 - a^7*b^2*e^14)) + 195*(7*B*b^20*d^5*e^8*abs(b) - 43*B*a*b^19*d^4*e^9*abs(b) + 8*A*b^20*d^
4*e^9*abs(b) + 102*B*a^2*b^18*d^3*e^10*abs(b) - 32*A*a*b^19*d^3*e^10*abs(b) - 118*B*a^3*b^17*d^2*e^11*abs(b) +
48*A*a^2*b^18*d^2*e^11*abs(b) + 67*B*a^4*b^16*d*e^12*abs(b) - 32*A*a^3*b^17*d*e^12*abs(b) - 15*B*a^5*b^15*e^1
3*abs(b) + 8*A*a^4*b^16*e^13*abs(b))/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10
+ 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a) + 715*(7*B*b^21*d^6
*e^7*abs(b) - 50*B*a*b^20*d^5*e^8*abs(b) + 8*A*b^21*d^5*e^8*abs(b) + 145*B*a^2*b^19*d^4*e^9*abs(b) - 40*A*a*b^
20*d^4*e^9*abs(b) - 220*B*a^3*b^18*d^3*e^10*abs(b) + 80*A*a^2*b^19*d^3*e^10*abs(b) + 185*B*a^4*b^17*d^2*e^11*a
bs(b) - 80*A*a^3*b^18*d^2*e^11*abs(b) - 82*B*a^5*b^16*d*e^12*abs(b) + 40*A*a^4*b^17*d*e^12*abs(b) + 15*B*a^6*b
^15*e^13*abs(b) - 8*A*a^5*b^16*e^13*abs(b))/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2*b^7*d^5*e^9 - 35*a^3*b^6*d
^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^14))*(b*x + a) - 6435*(B*a*
b^21*d^6*e^7*abs(b) - A*b^22*d^6*e^7*abs(b) - 6*B*a^2*b^20*d^5*e^8*abs(b) + 6*A*a*b^21*d^5*e^8*abs(b) + 15*B*a
^3*b^19*d^4*e^9*abs(b) - 15*A*a^2*b^20*d^4*e^9*abs(b) - 20*B*a^4*b^18*d^3*e^10*abs(b) + 20*A*a^3*b^19*d^3*e^10
*abs(b) + 15*B*a^5*b^17*d^2*e^11*abs(b) - 15*A*a^4*b^18*d^2*e^11*abs(b) - 6*B*a^6*b^16*d*e^12*abs(b) + 6*A*a^5
*b^17*d*e^12*abs(b) + B*a^7*b^15*e^13*abs(b) - A*a^6*b^16*e^13*abs(b))/(b^9*d^7*e^7 - 7*a*b^8*d^6*e^8 + 21*a^2
*b^7*d^5*e^9 - 35*a^3*b^6*d^4*e^10 + 35*a^4*b^5*d^3*e^11 - 21*a^5*b^4*d^2*e^12 + 7*a^6*b^3*d*e^13 - a^7*b^2*e^
14))*(b*x + a)^(7/2)/(b^2*d + (b*x + a)*b*e - a*b*e)^(15/2)

Mupad [B] (verification not implemented)

Time = 4.39 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.60 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (924\,B\,a^7\,d\,e^3+6006\,A\,a^7\,e^4-3780\,B\,a^6\,b\,d^2\,e^2-27720\,A\,a^6\,b\,d\,e^3+5460\,B\,a^5\,b^2\,d^3\,e+49140\,A\,a^5\,b^2\,d^2\,e^2-2860\,B\,a^4\,b^3\,d^4-40040\,A\,a^4\,b^3\,d^3\,e+12870\,A\,a^3\,b^4\,d^4\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^5}+\frac {x^2\,\sqrt {a+b\,x}\,\left (17010\,B\,a^6\,b\,e^4-77616\,B\,a^5\,b^2\,d\,e^3+8946\,A\,a^5\,b^2\,e^4+133620\,B\,a^4\,b^3\,d^2\,e^2-44520\,A\,a^4\,b^3\,d\,e^3-99840\,B\,a^3\,b^4\,d^3\,e+88140\,A\,a^3\,b^4\,d^2\,e^2+21450\,B\,a^2\,b^5\,d^4-85800\,A\,a^2\,b^5\,d^3\,e+38610\,A\,a\,b^6\,d^4\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^5}+\frac {x^3\,\sqrt {a+b\,x}\,\left (11130\,B\,a^5\,b^2\,e^4-55120\,B\,a^4\,b^3\,d\,e^3+70\,A\,a^4\,b^3\,e^4+107700\,B\,a^3\,b^4\,d^2\,e^2-600\,A\,a^3\,b^4\,d\,e^3-99840\,B\,a^2\,b^5\,d^3\,e+2340\,A\,a^2\,b^5\,d^2\,e^2+27170\,B\,a\,b^6\,d^4-5720\,A\,a\,b^6\,d^3\,e+12870\,A\,b^7\,d^4\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^5}+\frac {x\,\sqrt {a+b\,x}\,\left (6930\,B\,a^7\,e^4-28812\,B\,a^6\,b\,d\,e^3+14322\,A\,a^6\,b\,e^4+42840\,B\,a^5\,b^2\,d^2\,e^2-68040\,A\,a^5\,b^2\,d\,e^3-24180\,B\,a^4\,b^3\,d^3\,e+125580\,A\,a^4\,b^3\,d^2\,e^2+1430\,B\,a^3\,b^4\,d^4-108680\,A\,a^3\,b^4\,d^3\,e+38610\,A\,a^2\,b^5\,d^4\right )}{45045\,e^8\,{\left (a\,e-b\,d\right )}^5}+\frac {32\,b^6\,x^7\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-15\,B\,a\,e+7\,B\,b\,d\right )}{45045\,e^5\,{\left (a\,e-b\,d\right )}^5}-\frac {2\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-15\,B\,a\,e+7\,B\,b\,d\right )\,\left (a^3\,e^3-9\,a^2\,b\,d\,e^2+39\,a\,b^2\,d^2\,e-143\,b^3\,d^3\right )}{9009\,e^8\,{\left (a\,e-b\,d\right )}^5}-\frac {16\,b^5\,x^6\,\left (a\,e-15\,b\,d\right )\,\sqrt {a+b\,x}\,\left (8\,A\,b\,e-15\,B\,a\,e+7\,B\,b\,d\right )}{45045\,e^6\,{\left (a\,e-b\,d\right )}^5}+\frac {4\,b^4\,x^5\,\sqrt {a+b\,x}\,\left (a^2\,e^2-10\,a\,b\,d\,e+65\,b^2\,d^2\right )\,\left (8\,A\,b\,e-15\,B\,a\,e+7\,B\,b\,d\right )}{15015\,e^7\,{\left (a\,e-b\,d\right )}^5}\right )}{x^8+\frac {d^8}{e^8}+\frac {8\,d\,x^7}{e}+\frac {8\,d^7\,x}{e^7}+\frac {28\,d^2\,x^6}{e^2}+\frac {56\,d^3\,x^5}{e^3}+\frac {70\,d^4\,x^4}{e^4}+\frac {56\,d^5\,x^3}{e^5}+\frac {28\,d^6\,x^2}{e^6}} \]

[In]

int(((A + B*x)*(a + b*x)^(5/2))/(d + e*x)^(17/2),x)

[Out]

-((d + e*x)^(1/2)*(((a + b*x)^(1/2)*(6006*A*a^7*e^4 + 924*B*a^7*d*e^3 + 12870*A*a^3*b^4*d^4 - 2860*B*a^4*b^3*d
^4 - 40040*A*a^4*b^3*d^3*e + 5460*B*a^5*b^2*d^3*e - 3780*B*a^6*b*d^2*e^2 + 49140*A*a^5*b^2*d^2*e^2 - 27720*A*a
^6*b*d*e^3))/(45045*e^8*(a*e - b*d)^5) + (x^2*(a + b*x)^(1/2)*(38610*A*a*b^6*d^4 + 17010*B*a^6*b*e^4 + 8946*A*
a^5*b^2*e^4 + 21450*B*a^2*b^5*d^4 - 85800*A*a^2*b^5*d^3*e - 44520*A*a^4*b^3*d*e^3 - 99840*B*a^3*b^4*d^3*e - 77
616*B*a^5*b^2*d*e^3 + 88140*A*a^3*b^4*d^2*e^2 + 133620*B*a^4*b^3*d^2*e^2))/(45045*e^8*(a*e - b*d)^5) + (x^3*(a
+ b*x)^(1/2)*(12870*A*b^7*d^4 + 27170*B*a*b^6*d^4 + 70*A*a^4*b^3*e^4 + 11130*B*a^5*b^2*e^4 - 600*A*a^3*b^4*d*
e^3 - 99840*B*a^2*b^5*d^3*e - 55120*B*a^4*b^3*d*e^3 + 2340*A*a^2*b^5*d^2*e^2 + 107700*B*a^3*b^4*d^2*e^2 - 5720
*A*a*b^6*d^3*e))/(45045*e^8*(a*e - b*d)^5) + (x*(a + b*x)^(1/2)*(6930*B*a^7*e^4 + 14322*A*a^6*b*e^4 + 38610*A*
a^2*b^5*d^4 + 1430*B*a^3*b^4*d^4 - 108680*A*a^3*b^4*d^3*e - 68040*A*a^5*b^2*d*e^3 - 24180*B*a^4*b^3*d^3*e + 12
5580*A*a^4*b^3*d^2*e^2 + 42840*B*a^5*b^2*d^2*e^2 - 28812*B*a^6*b*d*e^3))/(45045*e^8*(a*e - b*d)^5) + (32*b^6*x
^7*(a + b*x)^(1/2)*(8*A*b*e - 15*B*a*e + 7*B*b*d))/(45045*e^5*(a*e - b*d)^5) - (2*b^3*x^4*(a + b*x)^(1/2)*(8*A
*b*e - 15*B*a*e + 7*B*b*d)*(a^3*e^3 - 143*b^3*d^3 + 39*a*b^2*d^2*e - 9*a^2*b*d*e^2))/(9009*e^8*(a*e - b*d)^5)
- (16*b^5*x^6*(a*e - 15*b*d)*(a + b*x)^(1/2)*(8*A*b*e - 15*B*a*e + 7*B*b*d))/(45045*e^6*(a*e - b*d)^5) + (4*b^
4*x^5*(a + b*x)^(1/2)*(a^2*e^2 + 65*b^2*d^2 - 10*a*b*d*e)*(8*A*b*e - 15*B*a*e + 7*B*b*d))/(15015*e^7*(a*e - b*
d)^5)))/(x^8 + d^8/e^8 + (8*d*x^7)/e + (8*d^7*x)/e^7 + (28*d^2*x^6)/e^2 + (56*d^3*x^5)/e^3 + (70*d^4*x^4)/e^4
+ (56*d^5*x^3)/e^5 + (28*d^6*x^2)/e^6)