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]

-1/9*(-A*e+B*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)/(e*x+d)^9+1/24*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^5*((b*
x+a)^2)^(1/2)/e/(-a*e+b*d)^2/(e*x+d)^8+1/84*b*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)
^3/(e*x+d)^7+1/504*b^2*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/e/(-a*e+b*d)^4/(e*x+d)^6

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Rubi [A]  time = 0.19, antiderivative size = 262, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.21, size = 468, normalized size = 1.79 \[ -\frac {\sqrt {(a+b x)^2} \left (7 a^5 e^5 (8 A e+B (d+9 e x))+5 a^4 b e^4 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+2 a^2 b^3 e^2 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+a b^4 e \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )\right )}{504 e^7 (a+b x) (d+e x)^9} \]

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]

-1/504*(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))))/(e^7*(a + b*x)*(d + e*x)^
9)

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fricas [B]  time = 0.59, size = 645, normalized size = 2.46 \[ -\frac {168 \, B b^{5} e^{6} x^{6} + 2 \, B b^{5} d^{6} + 56 \, A a^{5} e^{6} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 126 \, {\left (2 \, B b^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 126 \, {\left (2 \, B b^{5} d^{2} e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 84 \, {\left (2 \, B b^{5} d^{3} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 36 \, {\left (2 \, B b^{5} d^{4} e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 9 \, {\left (2 \, B b^{5} d^{5} e + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{504 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

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)

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giac [B]  time = 0.24, size = 918, normalized size = 3.50 \[ \text {result too large to display} \]

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

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maple [B]  time = 0.05, size = 688, normalized size = 2.63 \[ -\frac {\left (168 B \,b^{5} e^{6} x^{6}+126 A \,b^{5} e^{6} x^{5}+630 B a \,b^{4} e^{6} x^{5}+252 B \,b^{5} d \,e^{5} x^{5}+504 A a \,b^{4} e^{6} x^{4}+126 A \,b^{5} d \,e^{5} x^{4}+1008 B \,a^{2} b^{3} e^{6} x^{4}+630 B a \,b^{4} d \,e^{5} x^{4}+252 B \,b^{5} d^{2} e^{4} x^{4}+840 A \,a^{2} b^{3} e^{6} x^{3}+336 A a \,b^{4} d \,e^{5} x^{3}+84 A \,b^{5} d^{2} e^{4} x^{3}+840 B \,a^{3} b^{2} e^{6} x^{3}+672 B \,a^{2} b^{3} d \,e^{5} x^{3}+420 B a \,b^{4} d^{2} e^{4} x^{3}+168 B \,b^{5} d^{3} e^{3} x^{3}+720 A \,a^{3} b^{2} e^{6} x^{2}+360 A \,a^{2} b^{3} d \,e^{5} x^{2}+144 A a \,b^{4} d^{2} e^{4} x^{2}+36 A \,b^{5} d^{3} e^{3} x^{2}+360 B \,a^{4} b \,e^{6} x^{2}+360 B \,a^{3} b^{2} d \,e^{5} x^{2}+288 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+180 B a \,b^{4} d^{3} e^{3} x^{2}+72 B \,b^{5} d^{4} e^{2} x^{2}+315 A \,a^{4} b \,e^{6} x +180 A \,a^{3} b^{2} d \,e^{5} x +90 A \,a^{2} b^{3} d^{2} e^{4} x +36 A a \,b^{4} d^{3} e^{3} x +9 A \,b^{5} d^{4} e^{2} x +63 B \,a^{5} e^{6} x +90 B \,a^{4} b d \,e^{5} x +90 B \,a^{3} b^{2} d^{2} e^{4} x +72 B \,a^{2} b^{3} d^{3} e^{3} x +45 B a \,b^{4} d^{4} e^{2} x +18 B \,b^{5} d^{5} e x +56 A \,a^{5} e^{6}+35 A \,a^{4} b d \,e^{5}+20 A \,a^{3} b^{2} d^{2} e^{4}+10 A \,a^{2} b^{3} d^{3} e^{3}+4 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +7 B \,a^{5} d \,e^{5}+10 B \,a^{4} b \,d^{2} e^{4}+10 B \,a^{3} b^{2} d^{3} e^{3}+8 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +2 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (e x +d \right )^{9} \left (b x +a \right )^{5} e^{7}} \]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.64, size = 1489, normalized size = 5.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (((10*B*b^5*d^2 - 4*A*b^5*d*e + 5*A*a*b^4*e^2 + 10*B*a^2*b^3*e^2 - 20*B*a*b^4*d*e)/(5*e^7) - (d*((b^4*(A*b*e
+ 5*B*a*e - 4*B*b*d))/(5*e^6) - (B*b^5*d)/(5*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^
5) - (((A*b^5*e - 5*B*b^5*d + 5*B*a*b^4*e)/(4*e^7) - (B*b^5*d)/(4*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a +
b*x)*(d + e*x)^4) - (((A*a^5)/(9*e) - (d*((B*a^5 + 5*A*a^4*b)/(9*e) + (d*((d*((d*((d*((A*b^5 + 5*B*a*b^4)/(9*
e) - (B*b^5*d)/(9*e^2)))/e - (5*a*b^3*(A*b + 2*B*a))/(9*e)))/e + (10*a^2*b^2*(A*b + B*a))/(9*e)))/e - (5*a^3*b
*(2*A*b + B*a))/(9*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - (((6*A*b^5*d^2*e - 1
0*B*b^5*d^3 + 10*A*a^2*b^3*e^3 + 10*B*a^3*b^2*e^3 - 30*B*a^2*b^3*d*e^2 - 15*A*a*b^4*d*e^2 + 30*B*a*b^4*d^2*e)/
(6*e^7) - (d*((5*A*a*b^4*e^3 - 3*A*b^5*d*e^2 + 6*B*b^5*d^2*e + 10*B*a^2*b^3*e^3 - 15*B*a*b^4*d*e^2)/(6*e^7) -
(d*((b^4*(A*b*e + 5*B*a*e - 3*B*b*d))/(6*e^5) - (B*b^5*d)/(6*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((
a + b*x)*(d + e*x)^6) - (((B*a^5*e^5 - B*b^5*d^5 + 5*A*a^4*b*e^5 + A*b^5*d^4*e - 5*A*a*b^4*d^3*e^2 - 10*A*a^3*
b^2*d*e^4 + 10*A*a^2*b^3*d^2*e^3 - 10*B*a^2*b^3*d^3*e^2 + 10*B*a^3*b^2*d^2*e^3 + 5*B*a*b^4*d^4*e - 5*B*a^4*b*d
*e^4)/(8*e^7) - (d*((5*B*a^4*b*e^5 + B*b^5*d^4*e + 10*A*a^3*b^2*e^5 - A*b^5*d^3*e^2 + 5*A*a*b^4*d^2*e^3 - 10*A
*a^2*b^3*d*e^4 - 5*B*a*b^4*d^3*e^2 - 10*B*a^3*b^2*d*e^4 + 10*B*a^2*b^3*d^2*e^3)/(8*e^7) - (d*((10*A*a^2*b^3*e^
5 + 10*B*a^3*b^2*e^5 + A*b^5*d^2*e^3 - B*b^5*d^3*e^2 + 5*B*a*b^4*d^2*e^3 - 10*B*a^2*b^3*d*e^4 - 5*A*a*b^4*d*e^
4)/(8*e^7) - (d*((5*A*a*b^4*e^5 - A*b^5*d*e^4 + 10*B*a^2*b^3*e^5 + B*b^5*d^2*e^3 - 5*B*a*b^4*d*e^4)/(8*e^7) -
(d*((b^4*(A*b*e + 5*B*a*e - B*b*d))/(8*e^3) - (B*b^5*d)/(8*e^3)))/e))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/
2))/((a + b*x)*(d + e*x)^8) - (((5*B*b^5*d^4 + 5*B*a^4*b*e^4 - 4*A*b^5*d^3*e + 10*A*a^3*b^2*e^4 + 15*A*a*b^4*d
^2*e^2 - 20*A*a^2*b^3*d*e^3 - 20*B*a^3*b^2*d*e^3 + 30*B*a^2*b^3*d^2*e^2 - 20*B*a*b^4*d^3*e)/(7*e^7) - (d*((10*
A*a^2*b^3*e^4 - 4*B*b^5*d^3*e + 10*B*a^3*b^2*e^4 + 3*A*b^5*d^2*e^2 + 15*B*a*b^4*d^2*e^2 - 20*B*a^2*b^3*d*e^3 -
10*A*a*b^4*d*e^3)/(7*e^7) - (d*((5*A*a*b^4*e^4 - 2*A*b^5*d*e^3 + 10*B*a^2*b^3*e^4 + 3*B*b^5*d^2*e^2 - 10*B*a*
b^4*d*e^3)/(7*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 2*B*b*d))/(7*e^4) - (B*b^5*d)/(7*e^4)))/e))/e))/e)*(a^2 + b^2
*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*e^7*(a + b*x)*(d +
e*x)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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