3.1754 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=262 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{504 e (d+e x)^6 (b d-a e)^4}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{84 e (d+e x)^7 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{24 e (d+e x)^8 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e*(b*d - a*e)*(d + e*x)^9) + ((2*b*B*d + A*b*e - 3
*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(24*e*(b*d - a*e)^2*(d + e*x)^8) + (b*(2*b*B*d + A*b*e - 3*
a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(84*e*(b*d - a*e)^3*(d + e*x)^7) + (b^2*(2*b*B*d + A*b*e - 3
*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(504*e*(b*d - a*e)^4*(d + e*x)^6)

________________________________________________________________________________________

Rubi [A]  time = 0.192169, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {770, 78, 45, 37} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{504 e (d+e x)^6 (b d-a e)^4}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{84 e (d+e x)^7 (b d-a e)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{24 e (d+e x)^8 (b d-a e)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^10,x]

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e*(b*d - a*e)*(d + e*x)^9) + ((2*b*B*d + A*b*e - 3
*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(24*e*(b*d - a*e)^2*(d + e*x)^8) + (b*(2*b*B*d + A*b*e - 3*
a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(84*e*(b*d - a*e)^3*(d + e*x)^7) + (b^2*(2*b*B*d + A*b*e - 3
*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(504*e*(b*d - a*e)^4*(d + e*x)^6)

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 78

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] || LtQ
[p, n]))))

Rule 45

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 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]

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)^{10}} \, 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)^{10}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{(B d-A e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac{\left ((2 b B d+A b e-3 a B e) \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^9} \, dx}{3 b^4 e (b d-a e) \left (a b+b^2 x\right )}\\ &=-\frac{(B d-A e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac{(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac{\left ((2 b B d+A b e-3 a B e) \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^8} \, dx}{12 b^3 e (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=-\frac{(B d-A e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac{(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac{b (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{84 e (b d-a e)^3 (d+e x)^7}+\frac{\left ((2 b B d+A b e-3 a B e) \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{84 b^2 e (b d-a e)^3 \left (a b+b^2 x\right )}\\ &=-\frac{(B d-A e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e (b d-a e) (d+e x)^9}+\frac{(2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{24 e (b d-a e)^2 (d+e x)^8}+\frac{b (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{84 e (b d-a e)^3 (d+e x)^7}+\frac{b^2 (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{504 e (b d-a e)^4 (d+e x)^6}\\ \end{align*}

Mathematica [A]  time = 0.235517, size = 468, normalized size = 1.79 \[ -\frac{\sqrt{(a+b x)^2} \left (2 a^2 b^3 e^2 \left (5 A e \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (9 d^2 e x+d^3+36 d e^2 x^2+84 e^3 x^3\right )\right )+5 a^4 b e^4 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+7 a^5 e^5 (8 A e+B (d+9 e x))+a b^4 e \left (4 A e \left (36 d^2 e^2 x^2+9 d^3 e x+d^4+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )\right )+b^5 \left (A e \left (36 d^3 e^2 x^2+84 d^2 e^3 x^3+9 d^4 e x+d^5+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+9 d^5 e x+d^6+126 d e^5 x^5+84 e^6 x^6\right )\right )\right )}{504 e^7 (a+b x) (d+e x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^10,x]

[Out]

-(Sqrt[(a + b*x)^2]*(7*a^5*e^5*(8*A*e + B*(d + 9*e*x)) + 5*a^4*b*e^4*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x +
36*e^2*x^2)) + 10*a^3*b^2*e^3*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^
3*x^3)) + 2*a^2*b^3*e^2*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e
^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + a*b^4*e*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*
e^4*x^4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + b^5*(A*e*(
d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d^5*e*x + 36*d
^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6))))/(504*e^7*(a + b*x)*(d + e*x)^9)

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 688, normalized size = 2.6 \begin{align*} -{\frac{168\,B{x}^{6}{b}^{5}{e}^{6}+126\,A{x}^{5}{b}^{5}{e}^{6}+630\,B{x}^{5}a{b}^{4}{e}^{6}+252\,B{x}^{5}{b}^{5}d{e}^{5}+504\,A{x}^{4}a{b}^{4}{e}^{6}+126\,A{x}^{4}{b}^{5}d{e}^{5}+1008\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+630\,B{x}^{4}a{b}^{4}d{e}^{5}+252\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+840\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+336\,A{x}^{3}a{b}^{4}d{e}^{5}+84\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+840\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+672\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+420\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+168\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+720\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+360\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+144\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+36\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+360\,B{x}^{2}{a}^{4}b{e}^{6}+360\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+288\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+180\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+72\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+315\,Ax{a}^{4}b{e}^{6}+180\,Ax{a}^{3}{b}^{2}d{e}^{5}+90\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+36\,Axa{b}^{4}{d}^{3}{e}^{3}+9\,Ax{b}^{5}{d}^{4}{e}^{2}+63\,Bx{a}^{5}{e}^{6}+90\,Bx{a}^{4}bd{e}^{5}+90\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+72\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+45\,Bxa{b}^{4}{d}^{4}{e}^{2}+18\,Bx{b}^{5}{d}^{5}e+56\,A{a}^{5}{e}^{6}+35\,Ad{e}^{5}{a}^{4}b+20\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+10\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+4\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+7\,Bd{e}^{5}{a}^{5}+10\,B{a}^{4}b{d}^{2}{e}^{4}+10\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+8\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+2\,B{b}^{5}{d}^{6}}{504\,{e}^{7} \left ( ex+d \right ) ^{9} \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)^10,x)

[Out]

-1/504/e^7*(168*B*b^5*e^6*x^6+126*A*b^5*e^6*x^5+630*B*a*b^4*e^6*x^5+252*B*b^5*d*e^5*x^5+504*A*a*b^4*e^6*x^4+12
6*A*b^5*d*e^5*x^4+1008*B*a^2*b^3*e^6*x^4+630*B*a*b^4*d*e^5*x^4+252*B*b^5*d^2*e^4*x^4+840*A*a^2*b^3*e^6*x^3+336
*A*a*b^4*d*e^5*x^3+84*A*b^5*d^2*e^4*x^3+840*B*a^3*b^2*e^6*x^3+672*B*a^2*b^3*d*e^5*x^3+420*B*a*b^4*d^2*e^4*x^3+
168*B*b^5*d^3*e^3*x^3+720*A*a^3*b^2*e^6*x^2+360*A*a^2*b^3*d*e^5*x^2+144*A*a*b^4*d^2*e^4*x^2+36*A*b^5*d^3*e^3*x
^2+360*B*a^4*b*e^6*x^2+360*B*a^3*b^2*d*e^5*x^2+288*B*a^2*b^3*d^2*e^4*x^2+180*B*a*b^4*d^3*e^3*x^2+72*B*b^5*d^4*
e^2*x^2+315*A*a^4*b*e^6*x+180*A*a^3*b^2*d*e^5*x+90*A*a^2*b^3*d^2*e^4*x+36*A*a*b^4*d^3*e^3*x+9*A*b^5*d^4*e^2*x+
63*B*a^5*e^6*x+90*B*a^4*b*d*e^5*x+90*B*a^3*b^2*d^2*e^4*x+72*B*a^2*b^3*d^3*e^3*x+45*B*a*b^4*d^4*e^2*x+18*B*b^5*
d^5*e*x+56*A*a^5*e^6+35*A*a^4*b*d*e^5+20*A*a^3*b^2*d^2*e^4+10*A*a^2*b^3*d^3*e^3+4*A*a*b^4*d^4*e^2+A*b^5*d^5*e+
7*B*a^5*d*e^5+10*B*a^4*b*d^2*e^4+10*B*a^3*b^2*d^3*e^3+8*B*a^2*b^3*d^4*e^2+5*B*a*b^4*d^5*e+2*B*b^5*d^6)*((b*x+a
)^2)^(5/2)/(e*x+d)^9/(b*x+a)^5

________________________________________________________________________________________

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)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.64868, size = 1346, normalized size = 5.14 \begin{align*} -\frac{168 \, B b^{5} e^{6} x^{6} + 2 \, B b^{5} d^{6} + 56 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 10 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 126 \,{\left (2 \, B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 126 \,{\left (2 \, B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 84 \,{\left (2 \, B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 36 \,{\left (2 \, B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 10 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 9 \,{\left (2 \, B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 4 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{504 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} 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)^10,x, algorithm="fricas")

[Out]

-1/504*(168*B*b^5*e^6*x^6 + 2*B*b^5*d^6 + 56*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5*e + 4*(2*B*a^2*b^3 + A*a*b^4)
*d^4*e^2 + 10*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 10*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 7*(B*a^5 + 5*A*a^4*b)*d*e
^5 + 126*(2*B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 126*(2*B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*d*e^5 + 4*
(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 84*(2*B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5)*d^2*e^4 + 4*(2*B*a^2*b^3 + A*a*b^
4)*d*e^5 + 10*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 36*(2*B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 4*(2*B*a^
2*b^3 + A*a*b^4)*d^2*e^4 + 10*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 10*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 9*(2*B*b^5
*d^5*e + (5*B*a*b^4 + A*b^5)*d^4*e^2 + 4*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4
+ 10*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 7*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7
+ 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8*x + d^9
*e^7)

________________________________________________________________________________________

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)**10,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.18012, size = 1239, normalized size = 4.73 \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)^10,x, algorithm="giac")

[Out]

-1/504*(168*B*b^5*x^6*e^6*sgn(b*x + a) + 252*B*b^5*d*x^5*e^5*sgn(b*x + a) + 252*B*b^5*d^2*x^4*e^4*sgn(b*x + a)
+ 168*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 72*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 18*B*b^5*d^5*x*e*sgn(b*x + a) + 2*
B*b^5*d^6*sgn(b*x + a) + 630*B*a*b^4*x^5*e^6*sgn(b*x + a) + 126*A*b^5*x^5*e^6*sgn(b*x + a) + 630*B*a*b^4*d*x^4
*e^5*sgn(b*x + a) + 126*A*b^5*d*x^4*e^5*sgn(b*x + a) + 420*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) + 84*A*b^5*d^2*x^3
*e^4*sgn(b*x + a) + 180*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 36*A*b^5*d^3*x^2*e^3*sgn(b*x + a) + 45*B*a*b^4*d^4*
x*e^2*sgn(b*x + a) + 9*A*b^5*d^4*x*e^2*sgn(b*x + a) + 5*B*a*b^4*d^5*e*sgn(b*x + a) + A*b^5*d^5*e*sgn(b*x + a)
+ 1008*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 504*A*a*b^4*x^4*e^6*sgn(b*x + a) + 672*B*a^2*b^3*d*x^3*e^5*sgn(b*x + a
) + 336*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 288*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x + a) + 144*A*a*b^4*d^2*x^2*e^4*sgn(
b*x + a) + 72*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 36*A*a*b^4*d^3*x*e^3*sgn(b*x + a) + 8*B*a^2*b^3*d^4*e^2*sgn(b
*x + a) + 4*A*a*b^4*d^4*e^2*sgn(b*x + a) + 840*B*a^3*b^2*x^3*e^6*sgn(b*x + a) + 840*A*a^2*b^3*x^3*e^6*sgn(b*x
+ a) + 360*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 360*A*a^2*b^3*d*x^2*e^5*sgn(b*x + a) + 90*B*a^3*b^2*d^2*x*e^4*sg
n(b*x + a) + 90*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) + 10*A*a^2*b^3*d^3*e^3*sg
n(b*x + a) + 360*B*a^4*b*x^2*e^6*sgn(b*x + a) + 720*A*a^3*b^2*x^2*e^6*sgn(b*x + a) + 90*B*a^4*b*d*x*e^5*sgn(b*
x + a) + 180*A*a^3*b^2*d*x*e^5*sgn(b*x + a) + 10*B*a^4*b*d^2*e^4*sgn(b*x + a) + 20*A*a^3*b^2*d^2*e^4*sgn(b*x +
a) + 63*B*a^5*x*e^6*sgn(b*x + a) + 315*A*a^4*b*x*e^6*sgn(b*x + a) + 7*B*a^5*d*e^5*sgn(b*x + a) + 35*A*a^4*b*d
*e^5*sgn(b*x + a) + 56*A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^9