3.19.54 \(\int (A+B x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1854]

Optimal. Leaf size=452 \[ \frac {2 (b d-a e)^5 (B d-A e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)} \]

[Out]

2/3*(-a*e+b*d)^5*(-A*e+B*d)*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/5*(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b
*d)*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+10/7*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(7/2)*((b
*x+a)^2)^(1/2)/e^7/(b*x+a)-20/9*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^7/(b
*x+a)+10/11*b^3*(-a*e+b*d)*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/13*b^4*(-A*
b*e-5*B*a*e+6*B*b*d)*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/15*b^5*B*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/
e^7/(b*x+a)

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Rubi [A]
time = 0.14, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {784, 78} \begin {gather*} -\frac {20 b^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-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (2*(b*d - a*e)
^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (10*b*(b*d -
a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (20*b^2
*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (1
0*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x
)) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (
2*b^5*B*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(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*} \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (A+B x) \sqrt {d+e x} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (-B d+A e) \sqrt {d+e x}}{e^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{3/2}}{e^6}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{5/2}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{11/2}}{e^6}+\frac {b^{10} B (d+e x)^{13/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 (B d-A e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 490, normalized size = 1.08 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (3003 a^5 e^5 (-2 B d+5 A e+3 B e x)+2145 a^4 b e^4 \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-1430 a^3 b^2 e^3 \left (-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )+130 a^2 b^3 e^2 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-5 a b^4 e \left (-13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )+b^5 \left (5 A e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+B \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )\right )}{45045 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(3003*a^5*e^5*(-2*B*d + 5*A*e + 3*B*e*x) + 2145*a^4*b*e^4*(7*A*e*(-2*d +
3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 1430*a^3*b^2*e^3*(-3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(16
*d^3 - 24*d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3)) + 130*a^2*b^3*e^2*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2
+ 35*e^3*x^3) + B*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) - 5*a*b^4*e*(-13*A
*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 5*B*(256*d^5 - 384*d^4*e*x + 480*
d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^4*x^4 - 693*e^5*x^5)) + b^5*(5*A*e*(-256*d^5 + 384*d^4*e*x - 480*d^3*e
^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + B*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240
*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6))))/(45045*e^7*(a + b*x))

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Maple [A]
time = 0.98, size = 689, normalized size = 1.52 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*B*b^5*e^6*x^6+3465*A*b^5*e^6*x^5+17325*B*a*b^4*e^6*x^5-2772*B*b^5*d*e^5*x^5+20475*
A*a*b^4*e^6*x^4-3150*A*b^5*d*e^5*x^4+40950*B*a^2*b^3*e^6*x^4-15750*B*a*b^4*d*e^5*x^4+2520*B*b^5*d^2*e^4*x^4+50
050*A*a^2*b^3*e^6*x^3-18200*A*a*b^4*d*e^5*x^3+2800*A*b^5*d^2*e^4*x^3+50050*B*a^3*b^2*e^6*x^3-36400*B*a^2*b^3*d
*e^5*x^3+14000*B*a*b^4*d^2*e^4*x^3-2240*B*b^5*d^3*e^3*x^3+64350*A*a^3*b^2*e^6*x^2-42900*A*a^2*b^3*d*e^5*x^2+15
600*A*a*b^4*d^2*e^4*x^2-2400*A*b^5*d^3*e^3*x^2+32175*B*a^4*b*e^6*x^2-42900*B*a^3*b^2*d*e^5*x^2+31200*B*a^2*b^3
*d^2*e^4*x^2-12000*B*a*b^4*d^3*e^3*x^2+1920*B*b^5*d^4*e^2*x^2+45045*A*a^4*b*e^6*x-51480*A*a^3*b^2*d*e^5*x+3432
0*A*a^2*b^3*d^2*e^4*x-12480*A*a*b^4*d^3*e^3*x+1920*A*b^5*d^4*e^2*x+9009*B*a^5*e^6*x-25740*B*a^4*b*d*e^5*x+3432
0*B*a^3*b^2*d^2*e^4*x-24960*B*a^2*b^3*d^3*e^3*x+9600*B*a*b^4*d^4*e^2*x-1536*B*b^5*d^5*e*x+15015*A*a^5*e^6-3003
0*A*a^4*b*d*e^5+34320*A*a^3*b^2*d^2*e^4-22880*A*a^2*b^3*d^3*e^3+8320*A*a*b^4*d^4*e^2-1280*A*b^5*d^5*e-6006*B*a
^5*d*e^5+17160*B*a^4*b*d^2*e^4-22880*B*a^3*b^2*d^3*e^3+16640*B*a^2*b^3*d^4*e^2-6400*B*a*b^4*d^5*e+1024*B*b^5*d
^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [A]
time = 0.31, size = 710, normalized size = 1.57 \begin {gather*} \frac {2}{9009} \, {\left (693 \, b^{5} x^{6} e^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {x e + d} A e^{\left (-6\right )} + \frac {2}{45045} \, {\left (3003 \, b^{5} x^{7} e^{7} + 1024 \, b^{5} d^{7} - 6400 \, a b^{4} d^{6} e + 16640 \, a^{2} b^{3} d^{5} e^{2} - 22880 \, a^{3} b^{2} d^{4} e^{3} + 17160 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} + 231 \, {\left (b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{5} d^{2} e^{5} - 25 \, a b^{4} d e^{6} - 650 \, a^{2} b^{3} e^{7}\right )} x^{5} + 70 \, {\left (4 \, b^{5} d^{3} e^{4} - 25 \, a b^{4} d^{2} e^{5} + 65 \, a^{2} b^{3} d e^{6} + 715 \, a^{3} b^{2} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{5} d^{4} e^{3} - 400 \, a b^{4} d^{3} e^{4} + 1040 \, a^{2} b^{3} d^{2} e^{5} - 1430 \, a^{3} b^{2} d e^{6} - 6435 \, a^{4} b e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{5} e^{2} - 800 \, a b^{4} d^{4} e^{3} + 2080 \, a^{2} b^{3} d^{3} e^{4} - 2860 \, a^{3} b^{2} d^{2} e^{5} + 2145 \, a^{4} b d e^{6} + 3003 \, a^{5} e^{7}\right )} x^{2} - {\left (512 \, b^{5} d^{6} e - 3200 \, a b^{4} d^{5} e^{2} + 8320 \, a^{2} b^{3} d^{4} e^{3} - 11440 \, a^{3} b^{2} d^{3} e^{4} + 8580 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} \sqrt {x e + d} B e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/9009*(693*b^5*x^6*e^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
x*e + d)*A*e^(-6) + 2/45045*(3003*b^5*x^7*e^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b^3*d^5*e^2 - 2288
0*a^3*b^2*d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 231*(b^5*d*e^6 + 75*a*b^4*e^7)*x^6 - 63*(4*b^5*d^
2*e^5 - 25*a*b^4*d*e^6 - 650*a^2*b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*
a^3*b^2*e^7)*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 1430*a^3*b^2*d*e^6 - 6435*a^
4*b*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4*e^3 + 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4
*b*d*e^6 + 3003*a^5*e^7)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e^3 - 11440*a^3*b^2*d^3*
e^4 + 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt(x*e + d)*B*e^(-7)

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Fricas [A]
time = 3.26, size = 671, normalized size = 1.48 \begin {gather*} \frac {2}{45045} \, {\left (1024 \, B b^{5} d^{7} + {\left (3003 \, B b^{5} x^{7} + 15015 \, A a^{5} x + 3465 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + 20475 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + 50050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + 32175 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + 9009 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}\right )} e^{7} + {\left (231 \, B b^{5} d x^{6} + 15015 \, A a^{5} d + 315 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d x^{5} + 2275 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d x^{4} + 7150 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d x^{3} + 6435 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d x^{2} + 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d x\right )} e^{6} - 2 \, {\left (126 \, B b^{5} d^{2} x^{5} + 175 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} x^{4} + 1300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} x^{3} + 4290 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} x^{2} + 4290 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} x + 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2}\right )} e^{5} + 40 \, {\left (7 \, B b^{5} d^{3} x^{4} + 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} x^{3} + 78 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} x^{2} + 286 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} x + 429 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3}\right )} e^{4} - 160 \, {\left (2 \, B b^{5} d^{4} x^{3} + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} x^{2} + 26 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} x + 143 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4}\right )} e^{3} + 128 \, {\left (3 \, B b^{5} d^{5} x^{2} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} x + 65 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5}\right )} e^{2} - 256 \, {\left (2 \, B b^{5} d^{6} x + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(1024*B*b^5*d^7 + (3003*B*b^5*x^7 + 15015*A*a^5*x + 3465*(5*B*a*b^4 + A*b^5)*x^6 + 20475*(2*B*a^2*b^3
+ A*a*b^4)*x^5 + 50050*(B*a^3*b^2 + A*a^2*b^3)*x^4 + 32175*(B*a^4*b + 2*A*a^3*b^2)*x^3 + 9009*(B*a^5 + 5*A*a^4
*b)*x^2)*e^7 + (231*B*b^5*d*x^6 + 15015*A*a^5*d + 315*(5*B*a*b^4 + A*b^5)*d*x^5 + 2275*(2*B*a^2*b^3 + A*a*b^4)
*d*x^4 + 7150*(B*a^3*b^2 + A*a^2*b^3)*d*x^3 + 6435*(B*a^4*b + 2*A*a^3*b^2)*d*x^2 + 3003*(B*a^5 + 5*A*a^4*b)*d*
x)*e^6 - 2*(126*B*b^5*d^2*x^5 + 175*(5*B*a*b^4 + A*b^5)*d^2*x^4 + 1300*(2*B*a^2*b^3 + A*a*b^4)*d^2*x^3 + 4290*
(B*a^3*b^2 + A*a^2*b^3)*d^2*x^2 + 4290*(B*a^4*b + 2*A*a^3*b^2)*d^2*x + 3003*(B*a^5 + 5*A*a^4*b)*d^2)*e^5 + 40*
(7*B*b^5*d^3*x^4 + 10*(5*B*a*b^4 + A*b^5)*d^3*x^3 + 78*(2*B*a^2*b^3 + A*a*b^4)*d^3*x^2 + 286*(B*a^3*b^2 + A*a^
2*b^3)*d^3*x + 429*(B*a^4*b + 2*A*a^3*b^2)*d^3)*e^4 - 160*(2*B*b^5*d^4*x^3 + 3*(5*B*a*b^4 + A*b^5)*d^4*x^2 + 2
6*(2*B*a^2*b^3 + A*a*b^4)*d^4*x + 143*(B*a^3*b^2 + A*a^2*b^3)*d^4)*e^3 + 128*(3*B*b^5*d^5*x^2 + 5*(5*B*a*b^4 +
A*b^5)*d^5*x + 65*(2*B*a^2*b^3 + A*a*b^4)*d^5)*e^2 - 256*(2*B*b^5*d^6*x + 5*(5*B*a*b^4 + A*b^5)*d^6)*e)*sqrt(
x*e + d)*e^(-7)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)*sqrt(d + e*x)*((a + b*x)**2)**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1682 vs. \(2 (363) = 726\).
time = 1.80, size = 1682, normalized size = 3.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^5*d*e^(-1)*sgn(b*x + a) + 75075*((x*e + d)^(3/2) - 3*
sqrt(x*e + d)*d)*A*a^4*b*d*e^(-1)*sgn(b*x + a) + 15015*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e
+ d)*d^2)*B*a^4*b*d*e^(-2)*sgn(b*x + a) + 30030*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*
d^2)*A*a^3*b^2*d*e^(-2)*sgn(b*x + a) + 12870*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2 - 35*sqrt(x*e + d)*d^3)*B*a^3*b^2*d*e^(-3)*sgn(b*x + a) + 12870*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^2*b^3*d*e^(-3)*sgn(b*x + a) + 1430*(35*(x*e + d)^(9/2) - 18
0*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^2*b^3*d*e
^(-4)*sgn(b*x + a) + 715*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)
^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a*b^4*d*e^(-4)*sgn(b*x + a) + 325*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(
9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5
)*B*a*b^4*d*e^(-5)*sgn(b*x + a) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 -
1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*b^5*d*e^(-5)*sgn(b*x + a) + 15*
(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(
x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b^5*d*e^(-6)*sgn(b*x + a) + 3003*(3*
(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^5*e^(-1)*sgn(b*x + a) + 15015*(3*(x*e + d)^
(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4*b*e^(-1)*sgn(b*x + a) + 6435*(5*(x*e + d)^(7/2) - 2
1*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^4*b*e^(-2)*sgn(b*x + a) + 12870*(5*(x
*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^3*b^2*e^(-2)*sgn(b*x
+ a) + 1430*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 +
315*sqrt(x*e + d)*d^4)*B*a^3*b^2*e^(-3)*sgn(b*x + a) + 1430*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378
*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^2*b^3*e^(-3)*sgn(b*x + a) + 650*(6
3*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e +
d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^2*b^3*e^(-4)*sgn(b*x + a) + 325*(63*(x*e + d)^(11/2) - 385*(x*e + d)
^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d
^5)*A*a*b^4*e^(-4)*sgn(b*x + a) + 75*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^
2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B
*a*b^4*e^(-5)*sgn(b*x + a) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8
580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*b^5*
e^(-5)*sgn(b*x + a) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(
x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*
sqrt(x*e + d)*d^7)*B*b^5*e^(-6)*sgn(b*x + a) + 45045*sqrt(x*e + d)*A*a^5*d*sgn(b*x + a) + 15015*((x*e + d)^(3/
2) - 3*sqrt(x*e + d)*d)*A*a^5*sgn(b*x + a))*e^(-1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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