3.1854 \(\int (A+B x) \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.205869, antiderivative size = 452, normalized size of antiderivative = 1., 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 = {770, 77} \[ -\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}+\frac{2 b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \]

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 770

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]

Rule 77

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]))))

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*}

Mathematica [A]  time = 0.268735, size = 239, normalized size = 0.53 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (-3465 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+20475 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-50050 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+32175 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-9009 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+15015 (b d-a e)^5 (B d-A e)+3003 b^5 B (d+e x)^6\right )}{45045 e^7 (a+b x)} \]

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)*(15015*(b*d - a*e)^5*(B*d - A*e) - 9009*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e
- a*B*e)*(d + e*x) + 32175*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 50050*b^2*(b*d - a*e)^2*(
2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3 + 20475*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 3465*b^
4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 3003*b^5*B*(d + e*x)^6))/(45045*e^7*(a + b*x))

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Maple [A]  time = 0.009, size = 689, normalized size = 1.5 \begin{align*}{\frac{6006\,B{x}^{6}{b}^{5}{e}^{6}+6930\,A{x}^{5}{b}^{5}{e}^{6}+34650\,B{x}^{5}a{b}^{4}{e}^{6}-5544\,B{x}^{5}{b}^{5}d{e}^{5}+40950\,A{x}^{4}a{b}^{4}{e}^{6}-6300\,A{x}^{4}{b}^{5}d{e}^{5}+81900\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}-31500\,B{x}^{4}a{b}^{4}d{e}^{5}+5040\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+100100\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}-36400\,A{x}^{3}a{b}^{4}d{e}^{5}+5600\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+100100\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}-72800\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+28000\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}-4480\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+128700\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}-85800\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+31200\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}-4800\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+64350\,B{x}^{2}{a}^{4}b{e}^{6}-85800\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+62400\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}-24000\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+3840\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+90090\,Ax{a}^{4}b{e}^{6}-102960\,Ax{a}^{3}{b}^{2}d{e}^{5}+68640\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}-24960\,Axa{b}^{4}{d}^{3}{e}^{3}+3840\,Ax{b}^{5}{d}^{4}{e}^{2}+18018\,Bx{a}^{5}{e}^{6}-51480\,Bx{a}^{4}bd{e}^{5}+68640\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}-49920\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+19200\,Bxa{b}^{4}{d}^{4}{e}^{2}-3072\,Bx{b}^{5}{d}^{5}e+30030\,A{a}^{5}{e}^{6}-60060\,Ad{e}^{5}{a}^{4}b+68640\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}-45760\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+16640\,Aa{b}^{4}{d}^{4}{e}^{2}-2560\,A{b}^{5}{d}^{5}e-12012\,Bd{e}^{5}{a}^{5}+34320\,B{a}^{4}b{d}^{2}{e}^{4}-45760\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+33280\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}-12800\,Ba{b}^{4}{d}^{5}e+2048\,B{b}^{5}{d}^{6}}{45045\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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)

[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 [B]  time = 1.19466, size = 1026, normalized size = 2.27 \begin{align*} \text{result too large to display} \end{align*}

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*e^6*x^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(
e*x + d)*A/e^6 + 2/45045*(3003*b^5*e^7*x^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*(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(e*x + d)*B/e^7

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Fricas [B]  time = 1.33844, size = 1551, normalized size = 3.43 \begin{align*} \text{result too large to display} \end{align*}

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*(3003*B*b^5*e^7*x^7 + 1024*B*b^5*d^7 + 15015*A*a^5*d*e^6 - 1280*(5*B*a*b^4 + A*b^5)*d^6*e + 8320*(2*B*
a^2*b^3 + A*a*b^4)*d^5*e^2 - 22880*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^3 + 17160*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 6
006*(B*a^5 + 5*A*a^4*b)*d^2*e^5 + 231*(B*b^5*d*e^6 + 15*(5*B*a*b^4 + A*b^5)*e^7)*x^6 - 63*(4*B*b^5*d^2*e^5 - 5
*(5*B*a*b^4 + A*b^5)*d*e^6 - 325*(2*B*a^2*b^3 + A*a*b^4)*e^7)*x^5 + 35*(8*B*b^5*d^3*e^4 - 10*(5*B*a*b^4 + A*b^
5)*d^2*e^5 + 65*(2*B*a^2*b^3 + A*a*b^4)*d*e^6 + 1430*(B*a^3*b^2 + A*a^2*b^3)*e^7)*x^4 - 5*(64*B*b^5*d^4*e^3 -
80*(5*B*a*b^4 + A*b^5)*d^3*e^4 + 520*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^5 - 1430*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 - 64
35*(B*a^4*b + 2*A*a^3*b^2)*e^7)*x^3 + 3*(128*B*b^5*d^5*e^2 - 160*(5*B*a*b^4 + A*b^5)*d^4*e^3 + 1040*(2*B*a^2*b
^3 + A*a*b^4)*d^3*e^4 - 2860*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^5 + 2145*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 + 3003*(B*a^
5 + 5*A*a^4*b)*e^7)*x^2 - (512*B*b^5*d^6*e - 15015*A*a^5*e^7 - 640*(5*B*a*b^4 + A*b^5)*d^5*e^2 + 4160*(2*B*a^2
*b^3 + A*a*b^4)*d^4*e^3 - 11440*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^4 + 8580*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^5 - 3003*
(B*a^5 + 5*A*a^4*b)*d*e^6)*x)*sqrt(e*x + d)/e^7

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

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]

Timed out

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Giac [B]  time = 1.21372, size = 1027, normalized size = 2.27 \begin{align*} \text{result too large to display} \end{align*}

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*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^5*e^(-1)*sgn(b*x + a) + 15015*(3*(x*e + d)^(5/2) -
 5*(x*e + d)^(3/2)*d)*A*a^4*b*e^(-1)*sgn(b*x + a) + 2145*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e
+ d)^(3/2)*d^2)*B*a^4*b*e^(-2)*sgn(b*x + a) + 4290*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(
3/2)*d^2)*A*a^3*b^2*e^(-2)*sgn(b*x + a) + 1430*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/
2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^3*b^2*e^(-3)*sgn(b*x + a) + 1430*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/
2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a^2*b^3*e^(-3)*sgn(b*x + a) + 130*(315*(x*e + d)^(
11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^
4)*B*a^2*b^3*e^(-4)*sgn(b*x + a) + 65*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^
2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a*b^4*e^(-4)*sgn(b*x + a) + 25*(693*(x*e + d)^(13/2
) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4
 - 3003*(x*e + d)^(3/2)*d^5)*B*a*b^4*e^(-5)*sgn(b*x + a) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d +
 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*
A*b^5*e^(-5)*sgn(b*x + a) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 1
00100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)
*B*b^5*e^(-6)*sgn(b*x + a) + 15015*(x*e + d)^(3/2)*A*a^5*sgn(b*x + a))*e^(-1)