3.1758 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{14}} \, dx\)

Optimal. Leaf size=435 \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x) (d+e x)^9}+\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^{10}}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7} \]

[Out]

-((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^13) + ((b*d - a*e)^4*(6
*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)*(d + e*x)^12) - (5*b*(b*d - a*e)^3*
(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) + (b^2*(b*d - a*e)^
2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^10) - (5*b^3*(b*d - a*e)*(
3*b*B*d - A*b*e - 2*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(d + e*x)^9) + (b^4*(6*b*B*d - A*b*
e - 5*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)*(d + e*x)^8) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(7*e^7*(a + b*x)*(d + e*x)^7)

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Rubi [A]  time = 0.294193, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x) (d+e x)^9}+\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^{10}}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^14,x]

[Out]

-((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^13) + ((b*d - a*e)^4*(6
*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)*(d + e*x)^12) - (5*b*(b*d - a*e)^3*
(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) + (b^2*(b*d - a*e)^
2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^10) - (5*b^3*(b*d - a*e)*(
3*b*B*d - A*b*e - 2*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(d + e*x)^9) + (b^4*(6*b*B*d - A*b*
e - 5*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)*(d + e*x)^8) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(7*e^7*(a + b*x)*(d + e*x)^7)

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 \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{14}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{14}} \, 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)}{e^6 (d+e x)^{14}}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{13}}-\frac{5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^{12}}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^{11}}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^{10}}+\frac{b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^9}+\frac{b^{10} B}{e^6 (d+e x)^8}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{(b d-a e)^5 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac{(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{10}}-\frac{5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac{b^4 (6 b B d-A b e-5 a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.232856, size = 471, normalized size = 1.08 \[ -\frac{\sqrt{(a+b x)^2} \left (28 a^2 b^3 e^2 \left (9 A e \left (13 d^2 e x+d^3+78 d e^2 x^2+286 e^3 x^3\right )+4 B \left (78 d^2 e^2 x^2+13 d^3 e x+d^4+286 d e^3 x^3+715 e^4 x^4\right )\right )+84 a^3 b^2 e^3 \left (10 A e \left (d^2+13 d e x+78 e^2 x^2\right )+3 B \left (13 d^2 e x+d^3+78 d e^2 x^2+286 e^3 x^3\right )\right )+210 a^4 b e^4 \left (11 A e (d+13 e x)+2 B \left (d^2+13 d e x+78 e^2 x^2\right )\right )+462 a^5 e^5 (12 A e+B (d+13 e x))+7 a b^4 e \left (8 A e \left (78 d^2 e^2 x^2+13 d^3 e x+d^4+286 d e^3 x^3+715 e^4 x^4\right )+5 B \left (78 d^3 e^2 x^2+286 d^2 e^3 x^3+13 d^4 e x+d^5+715 d e^4 x^4+1287 e^5 x^5\right )\right )+b^5 \left (7 A e \left (78 d^3 e^2 x^2+286 d^2 e^3 x^3+13 d^4 e x+d^5+715 d e^4 x^4+1287 e^5 x^5\right )+6 B \left (78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+13 d^5 e x+d^6+1287 d e^5 x^5+1716 e^6 x^6\right )\right )\right )}{72072 e^7 (a+b x) (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^14,x]

[Out]

-(Sqrt[(a + b*x)^2]*(462*a^5*e^5*(12*A*e + B*(d + 13*e*x)) + 210*a^4*b*e^4*(11*A*e*(d + 13*e*x) + 2*B*(d^2 + 1
3*d*e*x + 78*e^2*x^2)) + 84*a^3*b^2*e^3*(10*A*e*(d^2 + 13*d*e*x + 78*e^2*x^2) + 3*B*(d^3 + 13*d^2*e*x + 78*d*e
^2*x^2 + 286*e^3*x^3)) + 28*a^2*b^3*e^2*(9*A*e*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 4*B*(d^4 + 13
*d^3*e*x + 78*d^2*e^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4)) + 7*a*b^4*e*(8*A*e*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^
2 + 286*d*e^3*x^3 + 715*e^4*x^4) + 5*B*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 +
1287*e^5*x^5)) + b^5*(7*A*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 + 1287*e^5*x^
5) + 6*B*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 + 1716*e^6*x^
6))))/(72072*e^7*(a + b*x)*(d + e*x)^13)

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Maple [A]  time = 0.015, size = 689, normalized size = 1.6 \begin{align*} -{\frac{10296\,B{x}^{6}{b}^{5}{e}^{6}+9009\,A{x}^{5}{b}^{5}{e}^{6}+45045\,B{x}^{5}a{b}^{4}{e}^{6}+7722\,B{x}^{5}{b}^{5}d{e}^{5}+40040\,A{x}^{4}a{b}^{4}{e}^{6}+5005\,A{x}^{4}{b}^{5}d{e}^{5}+80080\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+25025\,B{x}^{4}a{b}^{4}d{e}^{5}+4290\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+72072\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+16016\,A{x}^{3}a{b}^{4}d{e}^{5}+2002\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+72072\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+32032\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+10010\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+1716\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+65520\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+19656\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+4368\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+546\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+32760\,B{x}^{2}{a}^{4}b{e}^{6}+19656\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+8736\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+2730\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+468\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+30030\,Ax{a}^{4}b{e}^{6}+10920\,Ax{a}^{3}{b}^{2}d{e}^{5}+3276\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+728\,Axa{b}^{4}{d}^{3}{e}^{3}+91\,Ax{b}^{5}{d}^{4}{e}^{2}+6006\,Bx{a}^{5}{e}^{6}+5460\,Bx{a}^{4}bd{e}^{5}+3276\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+1456\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+455\,Bxa{b}^{4}{d}^{4}{e}^{2}+78\,Bx{b}^{5}{d}^{5}e+5544\,A{a}^{5}{e}^{6}+2310\,Ad{e}^{5}{a}^{4}b+840\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+252\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+56\,Aa{b}^{4}{d}^{4}{e}^{2}+7\,A{b}^{5}{d}^{5}e+462\,Bd{e}^{5}{a}^{5}+420\,B{a}^{4}b{d}^{2}{e}^{4}+252\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+112\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+35\,Ba{b}^{4}{d}^{5}e+6\,B{b}^{5}{d}^{6}}{72072\,{e}^{7} \left ( ex+d \right ) ^{13} \left ( bx+a \right ) ^{5}} \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)^14,x)

[Out]

-1/72072/e^7*(10296*B*b^5*e^6*x^6+9009*A*b^5*e^6*x^5+45045*B*a*b^4*e^6*x^5+7722*B*b^5*d*e^5*x^5+40040*A*a*b^4*
e^6*x^4+5005*A*b^5*d*e^5*x^4+80080*B*a^2*b^3*e^6*x^4+25025*B*a*b^4*d*e^5*x^4+4290*B*b^5*d^2*e^4*x^4+72072*A*a^
2*b^3*e^6*x^3+16016*A*a*b^4*d*e^5*x^3+2002*A*b^5*d^2*e^4*x^3+72072*B*a^3*b^2*e^6*x^3+32032*B*a^2*b^3*d*e^5*x^3
+10010*B*a*b^4*d^2*e^4*x^3+1716*B*b^5*d^3*e^3*x^3+65520*A*a^3*b^2*e^6*x^2+19656*A*a^2*b^3*d*e^5*x^2+4368*A*a*b
^4*d^2*e^4*x^2+546*A*b^5*d^3*e^3*x^2+32760*B*a^4*b*e^6*x^2+19656*B*a^3*b^2*d*e^5*x^2+8736*B*a^2*b^3*d^2*e^4*x^
2+2730*B*a*b^4*d^3*e^3*x^2+468*B*b^5*d^4*e^2*x^2+30030*A*a^4*b*e^6*x+10920*A*a^3*b^2*d*e^5*x+3276*A*a^2*b^3*d^
2*e^4*x+728*A*a*b^4*d^3*e^3*x+91*A*b^5*d^4*e^2*x+6006*B*a^5*e^6*x+5460*B*a^4*b*d*e^5*x+3276*B*a^3*b^2*d^2*e^4*
x+1456*B*a^2*b^3*d^3*e^3*x+455*B*a*b^4*d^4*e^2*x+78*B*b^5*d^5*e*x+5544*A*a^5*e^6+2310*A*a^4*b*d*e^5+840*A*a^3*
b^2*d^2*e^4+252*A*a^2*b^3*d^3*e^3+56*A*a*b^4*d^4*e^2+7*A*b^5*d^5*e+462*B*a^5*d*e^5+420*B*a^4*b*d^2*e^4+252*B*a
^3*b^2*d^3*e^3+112*B*a^2*b^3*d^4*e^2+35*B*a*b^4*d^5*e+6*B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^13/(b*x+a)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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)^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66209, size = 1520, normalized size = 3.49 \begin{align*} -\frac{10296 \, B b^{5} e^{6} x^{6} + 6 \, B b^{5} d^{6} + 5544 \, A a^{5} e^{6} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 420 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 462 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 1287 \,{\left (6 \, B b^{5} d e^{5} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 715 \,{\left (6 \, B b^{5} d^{2} e^{4} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 286 \,{\left (6 \, B b^{5} d^{3} e^{3} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 78 \,{\left (6 \, B b^{5} d^{4} e^{2} + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 420 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 13 \,{\left (6 \, B b^{5} d^{5} e + 7 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 56 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 252 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 420 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 462 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{72072 \,{\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \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)^14,x, algorithm="fricas")

[Out]

-1/72072*(10296*B*b^5*e^6*x^6 + 6*B*b^5*d^6 + 5544*A*a^5*e^6 + 7*(5*B*a*b^4 + A*b^5)*d^5*e + 56*(2*B*a^2*b^3 +
A*a*b^4)*d^4*e^2 + 252*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 420*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 462*(B*a^5 + 5
*A*a^4*b)*d*e^5 + 1287*(6*B*b^5*d*e^5 + 7*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 715*(6*B*b^5*d^2*e^4 + 7*(5*B*a*b^4 +
A*b^5)*d*e^5 + 56*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 286*(6*B*b^5*d^3*e^3 + 7*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 5
6*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 252*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 78*(6*B*b^5*d^4*e^2 + 7*(5*B*a*b^4 +
A*b^5)*d^3*e^3 + 56*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 252*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 420*(B*a^4*b + 2*A*a
^3*b^2)*e^6)*x^2 + 13*(6*B*b^5*d^5*e + 7*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 56*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 25
2*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 420*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 462*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^20*
x^13 + 13*d*e^19*x^12 + 78*d^2*e^18*x^11 + 286*d^3*e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^8 + 1716*d^6
*e^14*x^7 + 1716*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4 + 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 1
3*d^12*e^8*x + d^13*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)**14,x)

[Out]

Timed out

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Giac [B]  time = 1.19059, size = 1241, normalized size = 2.85 \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)^14,x, algorithm="giac")

[Out]

-1/72072*(10296*B*b^5*x^6*e^6*sgn(b*x + a) + 7722*B*b^5*d*x^5*e^5*sgn(b*x + a) + 4290*B*b^5*d^2*x^4*e^4*sgn(b*
x + a) + 1716*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 468*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 78*B*b^5*d^5*x*e*sgn(b*x +
a) + 6*B*b^5*d^6*sgn(b*x + a) + 45045*B*a*b^4*x^5*e^6*sgn(b*x + a) + 9009*A*b^5*x^5*e^6*sgn(b*x + a) + 25025*
B*a*b^4*d*x^4*e^5*sgn(b*x + a) + 5005*A*b^5*d*x^4*e^5*sgn(b*x + a) + 10010*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) +
2002*A*b^5*d^2*x^3*e^4*sgn(b*x + a) + 2730*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 546*A*b^5*d^3*x^2*e^3*sgn(b*x +
a) + 455*B*a*b^4*d^4*x*e^2*sgn(b*x + a) + 91*A*b^5*d^4*x*e^2*sgn(b*x + a) + 35*B*a*b^4*d^5*e*sgn(b*x + a) + 7*
A*b^5*d^5*e*sgn(b*x + a) + 80080*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 40040*A*a*b^4*x^4*e^6*sgn(b*x + a) + 32032*B
*a^2*b^3*d*x^3*e^5*sgn(b*x + a) + 16016*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 8736*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x +
a) + 4368*A*a*b^4*d^2*x^2*e^4*sgn(b*x + a) + 1456*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 728*A*a*b^4*d^3*x*e^3*sgn
(b*x + a) + 112*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 56*A*a*b^4*d^4*e^2*sgn(b*x + a) + 72072*B*a^3*b^2*x^3*e^6*sgn
(b*x + a) + 72072*A*a^2*b^3*x^3*e^6*sgn(b*x + a) + 19656*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 19656*A*a^2*b^3*d*
x^2*e^5*sgn(b*x + a) + 3276*B*a^3*b^2*d^2*x*e^4*sgn(b*x + a) + 3276*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 252*B*a
^3*b^2*d^3*e^3*sgn(b*x + a) + 252*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 32760*B*a^4*b*x^2*e^6*sgn(b*x + a) + 65520*
A*a^3*b^2*x^2*e^6*sgn(b*x + a) + 5460*B*a^4*b*d*x*e^5*sgn(b*x + a) + 10920*A*a^3*b^2*d*x*e^5*sgn(b*x + a) + 42
0*B*a^4*b*d^2*e^4*sgn(b*x + a) + 840*A*a^3*b^2*d^2*e^4*sgn(b*x + a) + 6006*B*a^5*x*e^6*sgn(b*x + a) + 30030*A*
a^4*b*x*e^6*sgn(b*x + a) + 462*B*a^5*d*e^5*sgn(b*x + a) + 2310*A*a^4*b*d*e^5*sgn(b*x + a) + 5544*A*a^5*e^6*sgn
(b*x + a))*e^(-7)/(x*e + d)^13