\(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\) [1844]

Optimal result

Integrand size = 35, antiderivative size = 308 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)} \]

[Out]

2/7*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-2/9*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b
*d)*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+6/11*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(11/2)*((b*x+
a)^2)^(1/2)/e^5/(b*x+a)-2/13*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+2/15*b^
3*B*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {784, 78} \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=-\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)} \]

[In]

Int[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) - (2*(b*d - a*e)
^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (6*b*(b*d -
a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) - (2*b^2*(4*
b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x)) + (2*b^3*B*(d + e*
x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^5*(a + b*x))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 784

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^{5/2} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^{5/2}}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{7/2}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{11/2}}{e^4}+\frac {b^6 B (d+e x)^{13/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.80 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (715 a^3 e^3 (-2 B d+9 A e+7 B e x)+195 a^2 b e^2 \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-15 a b^2 e \left (-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )+b^3 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )\right )}{45045 e^5 (a+b x)} \]

[In]

Integrate[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(715*a^3*e^3*(-2*B*d + 9*A*e + 7*B*e*x) + 195*a^2*b*e^2*(11*A*e*(-2*d + 7
*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 15*a*b^2*e*(-13*A*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B*(16*d^3
- 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + b^3*(15*A*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^
3) + B*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(45045*e^5*(a + b*x))

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.03

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 B \,b^{3} e^{4} x^{4}+3465 A \,b^{3} e^{4} x^{3}+10395 B a \,b^{2} e^{4} x^{3}-1848 B \,b^{3} d \,e^{3} x^{3}+12285 A a \,b^{2} e^{4} x^{2}-1890 A \,b^{3} d \,e^{3} x^{2}+12285 B \,a^{2} b \,e^{4} x^{2}-5670 B a \,b^{2} d \,e^{3} x^{2}+1008 B \,b^{3} d^{2} e^{2} x^{2}+15015 A \,a^{2} b \,e^{4} x -5460 A a \,b^{2} d \,e^{3} x +840 A \,b^{3} d^{2} e^{2} x +5005 B \,a^{3} e^{4} x -5460 B \,a^{2} b d \,e^{3} x +2520 B a \,b^{2} d^{2} e^{2} x -448 B \,b^{3} d^{3} e x +6435 A \,a^{3} e^{4}-4290 A \,a^{2} b d \,e^{3}+1560 A a \,b^{2} d^{2} e^{2}-240 A \,b^{3} d^{3} e -1430 B \,a^{3} d \,e^{3}+1560 B \,a^{2} b \,d^{2} e^{2}-720 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 e^{5} \left (b x +a \right )^{3}}\)	\(317\)
default	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 B \,b^{3} e^{4} x^{4}+3465 A \,b^{3} e^{4} x^{3}+10395 B a \,b^{2} e^{4} x^{3}-1848 B \,b^{3} d \,e^{3} x^{3}+12285 A a \,b^{2} e^{4} x^{2}-1890 A \,b^{3} d \,e^{3} x^{2}+12285 B \,a^{2} b \,e^{4} x^{2}-5670 B a \,b^{2} d \,e^{3} x^{2}+1008 B \,b^{3} d^{2} e^{2} x^{2}+15015 A \,a^{2} b \,e^{4} x -5460 A a \,b^{2} d \,e^{3} x +840 A \,b^{3} d^{2} e^{2} x +5005 B \,a^{3} e^{4} x -5460 B \,a^{2} b d \,e^{3} x +2520 B a \,b^{2} d^{2} e^{2} x -448 B \,b^{3} d^{3} e x +6435 A \,a^{3} e^{4}-4290 A \,a^{2} b d \,e^{3}+1560 A a \,b^{2} d^{2} e^{2}-240 A \,b^{3} d^{3} e -1430 B \,a^{3} d \,e^{3}+1560 B \,a^{2} b \,d^{2} e^{2}-720 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 e^{5} \left (b x +a \right )^{3}}\)	\(317\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{3} B \,e^{7} x^{7}+3465 A \,b^{3} e^{7} x^{6}+10395 B a \,b^{2} e^{7} x^{6}+7161 B \,b^{3} d \,e^{6} x^{6}+12285 A a \,b^{2} e^{7} x^{5}+8505 A \,b^{3} d \,e^{6} x^{5}+12285 B \,a^{2} b \,e^{7} x^{5}+25515 B a \,b^{2} d \,e^{6} x^{5}+4473 B \,b^{3} d^{2} e^{5} x^{5}+15015 A \,a^{2} b \,e^{7} x^{4}+31395 A a \,b^{2} d \,e^{6} x^{4}+5565 A \,b^{3} d^{2} e^{5} x^{4}+5005 B \,a^{3} e^{7} x^{4}+31395 B \,a^{2} b d \,e^{6} x^{4}+16695 B a \,b^{2} d^{2} e^{5} x^{4}+35 B \,b^{3} d^{3} e^{4} x^{4}+6435 A \,a^{3} e^{7} x^{3}+40755 A \,a^{2} b d \,e^{6} x^{3}+22035 A a \,b^{2} d^{2} e^{5} x^{3}+75 A \,b^{3} d^{3} e^{4} x^{3}+13585 B \,a^{3} d \,e^{6} x^{3}+22035 B \,a^{2} b \,d^{2} e^{5} x^{3}+225 B a \,b^{2} d^{3} e^{4} x^{3}-40 B \,b^{3} d^{4} e^{3} x^{3}+19305 A \,a^{3} d \,e^{6} x^{2}+32175 A \,a^{2} b \,d^{2} e^{5} x^{2}+585 A a \,b^{2} d^{3} e^{4} x^{2}-90 A \,b^{3} d^{4} e^{3} x^{2}+10725 B \,a^{3} d^{2} e^{5} x^{2}+585 B \,a^{2} b \,d^{3} e^{4} x^{2}-270 B a \,b^{2} d^{4} e^{3} x^{2}+48 B \,b^{3} d^{5} e^{2} x^{2}+19305 A \,a^{3} d^{2} e^{5} x +2145 A \,a^{2} b \,d^{3} e^{4} x -780 A a \,b^{2} d^{4} e^{3} x +120 A \,b^{3} d^{5} e^{2} x +715 B \,a^{3} d^{3} e^{4} x -780 B \,a^{2} b \,d^{4} e^{3} x +360 B a \,b^{2} d^{5} e^{2} x -64 B \,b^{3} d^{6} e x +6435 A \,a^{3} d^{3} e^{4}-4290 A \,a^{2} b \,d^{4} e^{3}+1560 A a \,b^{2} d^{5} e^{2}-240 A \,b^{3} d^{6} e -1430 B \,a^{3} d^{4} e^{3}+1560 B \,a^{2} b \,d^{5} e^{2}-720 B a \,b^{2} d^{6} e +128 B \,b^{3} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{5}}\)	\(685\)

[In]

int((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/45045*(e*x+d)^(7/2)*(3003*B*b^3*e^4*x^4+3465*A*b^3*e^4*x^3+10395*B*a*b^2*e^4*x^3-1848*B*b^3*d*e^3*x^3+12285*
A*a*b^2*e^4*x^2-1890*A*b^3*d*e^3*x^2+12285*B*a^2*b*e^4*x^2-5670*B*a*b^2*d*e^3*x^2+1008*B*b^3*d^2*e^2*x^2+15015
*A*a^2*b*e^4*x-5460*A*a*b^2*d*e^3*x+840*A*b^3*d^2*e^2*x+5005*B*a^3*e^4*x-5460*B*a^2*b*d*e^3*x+2520*B*a*b^2*d^2
*e^2*x-448*B*b^3*d^3*e*x+6435*A*a^3*e^4-4290*A*a^2*b*d*e^3+1560*A*a*b^2*d^2*e^2-240*A*b^3*d^3*e-1430*B*a^3*d*e
^3+1560*B*a^2*b*d^2*e^2-720*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (233) = 466\).

Time = 0.30 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (3003 \, B b^{3} e^{7} x^{7} + 128 \, B b^{3} d^{7} + 6435 \, A a^{3} d^{3} e^{4} - 240 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e + 1560 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{2} - 1430 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{3} + 231 \, {\left (31 \, B b^{3} d e^{6} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{7}\right )} x^{6} + 63 \, {\left (71 \, B b^{3} d^{2} e^{5} + 135 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{6} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} e^{7}\right )} x^{5} + 35 \, {\left (B b^{3} d^{3} e^{4} + 159 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{5} + 897 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{6} + 143 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{4} e^{3} - 1287 \, A a^{3} e^{7} - 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{4} - 4407 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{5} - 2717 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{5} e^{2} + 6435 \, A a^{3} d e^{6} - 30 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{3} + 195 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{4} + 3575 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{6} e - 19305 \, A a^{3} d^{2} e^{5} - 120 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{2} + 780 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{3} - 715 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]

[In]

integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^3*e^7*x^7 + 128*B*b^3*d^7 + 6435*A*a^3*d^3*e^4 - 240*(3*B*a*b^2 + A*b^3)*d^6*e + 1560*(B*a^2
*b + A*a*b^2)*d^5*e^2 - 1430*(B*a^3 + 3*A*a^2*b)*d^4*e^3 + 231*(31*B*b^3*d*e^6 + 15*(3*B*a*b^2 + A*b^3)*e^7)*x
^6 + 63*(71*B*b^3*d^2*e^5 + 135*(3*B*a*b^2 + A*b^3)*d*e^6 + 195*(B*a^2*b + A*a*b^2)*e^7)*x^5 + 35*(B*b^3*d^3*e
^4 + 159*(3*B*a*b^2 + A*b^3)*d^2*e^5 + 897*(B*a^2*b + A*a*b^2)*d*e^6 + 143*(B*a^3 + 3*A*a^2*b)*e^7)*x^4 - 5*(8
*B*b^3*d^4*e^3 - 1287*A*a^3*e^7 - 15*(3*B*a*b^2 + A*b^3)*d^3*e^4 - 4407*(B*a^2*b + A*a*b^2)*d^2*e^5 - 2717*(B*
a^3 + 3*A*a^2*b)*d*e^6)*x^3 + 3*(16*B*b^3*d^5*e^2 + 6435*A*a^3*d*e^6 - 30*(3*B*a*b^2 + A*b^3)*d^4*e^3 + 195*(B
*a^2*b + A*a*b^2)*d^3*e^4 + 3575*(B*a^3 + 3*A*a^2*b)*d^2*e^5)*x^2 - (64*B*b^3*d^6*e - 19305*A*a^3*d^2*e^5 - 12
0*(3*B*a*b^2 + A*b^3)*d^5*e^2 + 780*(B*a^2*b + A*a*b^2)*d^4*e^3 - 715*(B*a^3 + 3*A*a^2*b)*d^3*e^4)*x)*sqrt(e*x
+ d)/e^5

Sympy [F(-1)]

Timed out. \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (233) = 466\).

Time = 0.22 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.92 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \, {\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} + {\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} A}{3003 \, e^{4}} + \frac {2 \, {\left (3003 \, b^{3} e^{7} x^{7} + 128 \, b^{3} d^{7} - 720 \, a b^{2} d^{6} e + 1560 \, a^{2} b d^{5} e^{2} - 1430 \, a^{3} d^{4} e^{3} + 231 \, {\left (31 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{3} d^{2} e^{5} + 405 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 35 \, {\left (b^{3} d^{3} e^{4} + 477 \, a b^{2} d^{2} e^{5} + 897 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{4} e^{3} - 45 \, a b^{2} d^{3} e^{4} - 4407 \, a^{2} b d^{2} e^{5} - 2717 \, a^{3} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{5} e^{2} - 90 \, a b^{2} d^{4} e^{3} + 195 \, a^{2} b d^{3} e^{4} + 3575 \, a^{3} d^{2} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{6} e - 360 \, a b^{2} d^{5} e^{2} + 780 \, a^{2} b d^{4} e^{3} - 715 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d} B}{45045 \, e^{5}} \]

[In]

integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

2/3003*(231*b^3*e^6*x^6 - 16*b^3*d^6 + 104*a*b^2*d^5*e - 286*a^2*b*d^4*e^2 + 429*a^3*d^3*e^3 + 63*(9*b^3*d*e^5
+ 13*a*b^2*e^6)*x^5 + 7*(53*b^3*d^2*e^4 + 299*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 + (5*b^3*d^3*e^3 + 1469*a*b^2*
d^2*e^4 + 2717*a^2*b*d*e^5 + 429*a^3*e^6)*x^3 - 3*(2*b^3*d^4*e^2 - 13*a*b^2*d^3*e^3 - 715*a^2*b*d^2*e^4 - 429*
a^3*d*e^5)*x^2 + (8*b^3*d^5*e - 52*a*b^2*d^4*e^2 + 143*a^2*b*d^3*e^3 + 1287*a^3*d^2*e^4)*x)*sqrt(e*x + d)*A/e^
4 + 2/45045*(3003*b^3*e^7*x^7 + 128*b^3*d^7 - 720*a*b^2*d^6*e + 1560*a^2*b*d^5*e^2 - 1430*a^3*d^4*e^3 + 231*(3
1*b^3*d*e^6 + 45*a*b^2*e^7)*x^6 + 63*(71*b^3*d^2*e^5 + 405*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 35*(b^3*d^3*e^4
+ 477*a*b^2*d^2*e^5 + 897*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 - 5*(8*b^3*d^4*e^3 - 45*a*b^2*d^3*e^4 - 4407*a^2*b*d^
2*e^5 - 2717*a^3*d*e^6)*x^3 + 3*(16*b^3*d^5*e^2 - 90*a*b^2*d^4*e^3 + 195*a^2*b*d^3*e^4 + 3575*a^3*d^2*e^5)*x^2
- (64*b^3*d^6*e - 360*a*b^2*d^5*e^2 + 780*a^2*b*d^4*e^3 - 715*a^3*d^3*e^4)*x)*sqrt(e*x + d)*B/e^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2139 vs. \(2 (233) = 466\).

Time = 0.35 (sec) , antiderivative size = 2139, normalized size of antiderivative = 6.94 \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(e*x + d)*A*a^3*d^3*sgn(b*x + a) + 45045*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^3*d^2*sg
n(b*x + a) + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^3*d^3*sgn(b*x + a)/e + 45045*((e*x + d)^(3/2) - 3
*sqrt(e*x + d)*d)*A*a^2*b*d^3*sgn(b*x + a)/e + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x +
d)*d^2)*A*a^3*d*sgn(b*x + a) + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^2*b*
d^3*sgn(b*x + a)/e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a*b^2*d^3*sgn(
b*x + a)/e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^3*d^2*sgn(b*x + a)/e
+ 27027*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^2*b*d^2*sgn(b*x + a)/e + 1287*(
5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a^3*sgn(b*x + a) +
3861*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*a*b^2*d^3*s
gn(b*x + a)/e^3 + 1287*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d
^3)*A*b^3*d^3*sgn(b*x + a)/e^3 + 11583*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35
*sqrt(e*x + d)*d^3)*B*a^2*b*d^2*sgn(b*x + a)/e^2 + 11583*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x +
d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a*b^2*d^2*sgn(b*x + a)/e^2 + 3861*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5
/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*a^3*d*sgn(b*x + a)/e + 11583*(5*(e*x + d)^(7/2) - 21*
(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*a^2*b*d*sgn(b*x + a)/e + 143*(35*(e*x + d
)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*B
*b^3*d^3*sgn(b*x + a)/e^4 + 1287*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(
e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*B*a*b^2*d^2*sgn(b*x + a)/e^3 + 429*(35*(e*x + d)^(9/2) - 180*(e*x
+ d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*A*b^3*d^2*sgn(b*x +
a)/e^3 + 1287*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3
+ 315*sqrt(e*x + d)*d^4)*B*a^2*b*d*sgn(b*x + a)/e^2 + 1287*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(
e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*A*a*b^2*d*sgn(b*x + a)/e^2 + 143*(35*(e*
x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d
^4)*B*a^3*sgn(b*x + a)/e + 429*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*
x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*A*a^2*b*sgn(b*x + a)/e + 195*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9
/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)
*B*b^3*d^2*sgn(b*x + a)/e^4 + 585*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 138
6*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*B*a*b^2*d*sgn(b*x + a)/e^3 + 195*(63
*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d
)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*A*b^3*d*sgn(b*x + a)/e^3 + 195*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)
*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*B*
a^2*b*sgn(b*x + a)/e^2 + 195*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*
x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*A*a*b^2*sgn(b*x + a)/e^2 + 45*(231*(e*x +
d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5
/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*B*b^3*d*sgn(b*x + a)/e^4 + 45*(231*(e*x + d)^(13/
2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4
- 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*B*a*b^2*sgn(b*x + a)/e^3 + 15*(231*(e*x + d)^(13/2) - 163
8*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(
e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*A*b^3*sgn(b*x + a)/e^3 + 7*(429*(e*x + d)^(15/2) - 3465*(e*x + d)
^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d^3 + 32175*(e*x + d)^(7/2)*d^4 - 27027*(e*x +
d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d^7)*B*b^3*sgn(b*x + a)/e^4)/e

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]

[In]

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

[Out]

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