3.16.35 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=438 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{10 e^7 (a+b x) (d+e x)^{10}}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{5 e^7 (a+b x) (d+e x)^5}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{6 e^7 (a+b x) (d+e x)^6}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.39, antiderivative size = 438, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)}{5 e^7 (a+b x) (d+e x)^5}-\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)}{6 e^7 (a+b x) (d+e x)^6}+\frac {10 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)}{7 e^7 (a+b x) (d+e x)^7}-\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)}{8 e^7 (a+b x) (d+e x)^8}+\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)}{9 e^7 (a+b x) (d+e x)^9}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{10 e^7 (a+b x) (d+e x)^{10}}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^11,x]

[Out]

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

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]))))

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)^{11}} \, 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)^{11}} \, 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)^{11}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{10}}-\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)^9}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^8}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^7}+\frac {b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^6}+\frac {b^{10} B}{e^6 (d+e x)^5}\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}}{10 e^7 (a+b x) (d+e x)^{10}}+\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}}{9 e^7 (a+b x) (d+e x)^9}-\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}}{8 e^7 (a+b x) (d+e x)^8}+\frac {10 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}}{7 e^7 (a+b x) (d+e x)^7}-\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}}{6 e^7 (a+b x) (d+e x)^6}+\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}}{5 e^7 (a+b x) (d+e x)^5}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 468, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (28 a^5 e^5 (9 A e+B (d+10 e x))+35 a^4 b e^4 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+10 a b^4 e \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+b^5 \left (2 A e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 B \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )\right )}{2520 e^7 (a+b x) (d+e x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^11,x]

[Out]

-1/2520*(Sqrt[(a + b*x)^2]*(28*a^5*e^5*(9*A*e + B*(d + 10*e*x)) + 35*a^4*b*e^4*(4*A*e*(d + 10*e*x) + B*(d^2 +
10*d*e*x + 45*e^2*x^2)) + 10*a^3*b^2*e^3*(7*A*e*(d^2 + 10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e
^2*x^2 + 120*e^3*x^3)) + 10*a^2*b^3*e^2*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B*(d^4 + 10
*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 10*a*b^4*e*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2
+ 120*d*e^3*x^3 + 210*e^4*x^4) + B*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252
*e^5*x^5)) + b^5*(2*A*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) +
3*B*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6))))/(
e^7*(a + b*x)*(d + e*x)^10)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 180.07, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^11,x]

[Out]

$Aborted

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 662, normalized size = 1.51 \begin {gather*} -\frac {630 \, B b^{5} e^{6} x^{6} + 3 \, B b^{5} d^{6} + 252 \, A a^{5} e^{6} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 28 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 252 \, {\left (3 \, B b^{5} d e^{5} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 210 \, {\left (3 \, B b^{5} d^{2} e^{4} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 120 \, {\left (3 \, B b^{5} d^{3} e^{3} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 45 \, {\left (3 \, B b^{5} d^{4} e^{2} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 10 \, {\left (3 \, B b^{5} d^{5} e + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 35 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 28 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{2520 \, {\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{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)^11,x, algorithm="fricas")

[Out]

-1/2520*(630*B*b^5*e^6*x^6 + 3*B*b^5*d^6 + 252*A*a^5*e^6 + 2*(5*B*a*b^4 + A*b^5)*d^5*e + 10*(2*B*a^2*b^3 + A*a
*b^4)*d^4*e^2 + 30*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 35*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 28*(B*a^5 + 5*A*a^4*
b)*d*e^5 + 252*(3*B*b^5*d*e^5 + 2*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 210*(3*B*b^5*d^2*e^4 + 2*(5*B*a*b^4 + A*b^5)*
d*e^5 + 10*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 120*(3*B*b^5*d^3*e^3 + 2*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 10*(2*B*a
^2*b^3 + A*a*b^4)*d*e^5 + 30*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 45*(3*B*b^5*d^4*e^2 + 2*(5*B*a*b^4 + A*b^5)*d^
3*e^3 + 10*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 30*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 35*(B*a^4*b + 2*A*a^3*b^2)*e^6
)*x^2 + 10*(3*B*b^5*d^5*e + 2*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 10*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 30*(B*a^3*b^2
+ A*a^2*b^3)*d^2*e^4 + 35*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 28*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^17*x^10 + 10*d*e^
16*x^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7
*e^10*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)

________________________________________________________________________________________

giac [B]  time = 0.22, size = 919, normalized size = 2.10

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)^11,x, algorithm="giac")

[Out]

-1/2520*(630*B*b^5*x^6*e^6*sgn(b*x + a) + 756*B*b^5*d*x^5*e^5*sgn(b*x + a) + 630*B*b^5*d^2*x^4*e^4*sgn(b*x + a
) + 360*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 135*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 30*B*b^5*d^5*x*e*sgn(b*x + a) +
3*B*b^5*d^6*sgn(b*x + a) + 2520*B*a*b^4*x^5*e^6*sgn(b*x + a) + 504*A*b^5*x^5*e^6*sgn(b*x + a) + 2100*B*a*b^4*d
*x^4*e^5*sgn(b*x + a) + 420*A*b^5*d*x^4*e^5*sgn(b*x + a) + 1200*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) + 240*A*b^5*d
^2*x^3*e^4*sgn(b*x + a) + 450*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 90*A*b^5*d^3*x^2*e^3*sgn(b*x + a) + 100*B*a*b
^4*d^4*x*e^2*sgn(b*x + a) + 20*A*b^5*d^4*x*e^2*sgn(b*x + a) + 10*B*a*b^4*d^5*e*sgn(b*x + a) + 2*A*b^5*d^5*e*sg
n(b*x + a) + 4200*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 2100*A*a*b^4*x^4*e^6*sgn(b*x + a) + 2400*B*a^2*b^3*d*x^3*e^
5*sgn(b*x + a) + 1200*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 900*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x + a) + 450*A*a*b^4*d^
2*x^2*e^4*sgn(b*x + a) + 200*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 100*A*a*b^4*d^3*x*e^3*sgn(b*x + a) + 20*B*a^2*
b^3*d^4*e^2*sgn(b*x + a) + 10*A*a*b^4*d^4*e^2*sgn(b*x + a) + 3600*B*a^3*b^2*x^3*e^6*sgn(b*x + a) + 3600*A*a^2*
b^3*x^3*e^6*sgn(b*x + a) + 1350*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 1350*A*a^2*b^3*d*x^2*e^5*sgn(b*x + a) + 300
*B*a^3*b^2*d^2*x*e^4*sgn(b*x + a) + 300*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 30*B*a^3*b^2*d^3*e^3*sgn(b*x + a) +
30*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 1575*B*a^4*b*x^2*e^6*sgn(b*x + a) + 3150*A*a^3*b^2*x^2*e^6*sgn(b*x + a) +
350*B*a^4*b*d*x*e^5*sgn(b*x + a) + 700*A*a^3*b^2*d*x*e^5*sgn(b*x + a) + 35*B*a^4*b*d^2*e^4*sgn(b*x + a) + 70*
A*a^3*b^2*d^2*e^4*sgn(b*x + a) + 280*B*a^5*x*e^6*sgn(b*x + a) + 1400*A*a^4*b*x*e^6*sgn(b*x + a) + 28*B*a^5*d*e
^5*sgn(b*x + a) + 140*A*a^4*b*d*e^5*sgn(b*x + a) + 252*A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^10

________________________________________________________________________________________

maple [A]  time = 0.06, size = 689, normalized size = 1.57 \begin {gather*} -\frac {\left (630 B \,b^{5} e^{6} x^{6}+504 A \,b^{5} e^{6} x^{5}+2520 B a \,b^{4} e^{6} x^{5}+756 B \,b^{5} d \,e^{5} x^{5}+2100 A a \,b^{4} e^{6} x^{4}+420 A \,b^{5} d \,e^{5} x^{4}+4200 B \,a^{2} b^{3} e^{6} x^{4}+2100 B a \,b^{4} d \,e^{5} x^{4}+630 B \,b^{5} d^{2} e^{4} x^{4}+3600 A \,a^{2} b^{3} e^{6} x^{3}+1200 A a \,b^{4} d \,e^{5} x^{3}+240 A \,b^{5} d^{2} e^{4} x^{3}+3600 B \,a^{3} b^{2} e^{6} x^{3}+2400 B \,a^{2} b^{3} d \,e^{5} x^{3}+1200 B a \,b^{4} d^{2} e^{4} x^{3}+360 B \,b^{5} d^{3} e^{3} x^{3}+3150 A \,a^{3} b^{2} e^{6} x^{2}+1350 A \,a^{2} b^{3} d \,e^{5} x^{2}+450 A a \,b^{4} d^{2} e^{4} x^{2}+90 A \,b^{5} d^{3} e^{3} x^{2}+1575 B \,a^{4} b \,e^{6} x^{2}+1350 B \,a^{3} b^{2} d \,e^{5} x^{2}+900 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+450 B a \,b^{4} d^{3} e^{3} x^{2}+135 B \,b^{5} d^{4} e^{2} x^{2}+1400 A \,a^{4} b \,e^{6} x +700 A \,a^{3} b^{2} d \,e^{5} x +300 A \,a^{2} b^{3} d^{2} e^{4} x +100 A a \,b^{4} d^{3} e^{3} x +20 A \,b^{5} d^{4} e^{2} x +280 B \,a^{5} e^{6} x +350 B \,a^{4} b d \,e^{5} x +300 B \,a^{3} b^{2} d^{2} e^{4} x +200 B \,a^{2} b^{3} d^{3} e^{3} x +100 B a \,b^{4} d^{4} e^{2} x +30 B \,b^{5} d^{5} e x +252 A \,a^{5} e^{6}+140 A \,a^{4} b d \,e^{5}+70 A \,a^{3} b^{2} d^{2} e^{4}+30 A \,a^{2} b^{3} d^{3} e^{3}+10 A a \,b^{4} d^{4} e^{2}+2 A \,b^{5} d^{5} e +28 B \,a^{5} d \,e^{5}+35 B \,a^{4} b \,d^{2} e^{4}+30 B \,a^{3} b^{2} d^{3} e^{3}+20 B \,a^{2} b^{3} d^{4} e^{2}+10 B a \,b^{4} d^{5} e +3 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2520 \left (e x +d \right )^{10} \left (b x +a \right )^{5} e^{7}} \end {gather*}

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)^11,x)

[Out]

-1/2520/e^7*(630*B*b^5*e^6*x^6+504*A*b^5*e^6*x^5+2520*B*a*b^4*e^6*x^5+756*B*b^5*d*e^5*x^5+2100*A*a*b^4*e^6*x^4
+420*A*b^5*d*e^5*x^4+4200*B*a^2*b^3*e^6*x^4+2100*B*a*b^4*d*e^5*x^4+630*B*b^5*d^2*e^4*x^4+3600*A*a^2*b^3*e^6*x^
3+1200*A*a*b^4*d*e^5*x^3+240*A*b^5*d^2*e^4*x^3+3600*B*a^3*b^2*e^6*x^3+2400*B*a^2*b^3*d*e^5*x^3+1200*B*a*b^4*d^
2*e^4*x^3+360*B*b^5*d^3*e^3*x^3+3150*A*a^3*b^2*e^6*x^2+1350*A*a^2*b^3*d*e^5*x^2+450*A*a*b^4*d^2*e^4*x^2+90*A*b
^5*d^3*e^3*x^2+1575*B*a^4*b*e^6*x^2+1350*B*a^3*b^2*d*e^5*x^2+900*B*a^2*b^3*d^2*e^4*x^2+450*B*a*b^4*d^3*e^3*x^2
+135*B*b^5*d^4*e^2*x^2+1400*A*a^4*b*e^6*x+700*A*a^3*b^2*d*e^5*x+300*A*a^2*b^3*d^2*e^4*x+100*A*a*b^4*d^3*e^3*x+
20*A*b^5*d^4*e^2*x+280*B*a^5*e^6*x+350*B*a^4*b*d*e^5*x+300*B*a^3*b^2*d^2*e^4*x+200*B*a^2*b^3*d^3*e^3*x+100*B*a
*b^4*d^4*e^2*x+30*B*b^5*d^5*e*x+252*A*a^5*e^6+140*A*a^4*b*d*e^5+70*A*a^3*b^2*d^2*e^4+30*A*a^2*b^3*d^3*e^3+10*A
*a*b^4*d^4*e^2+2*A*b^5*d^5*e+28*B*a^5*d*e^5+35*B*a^4*b*d^2*e^4+30*B*a^3*b^2*d^3*e^3+20*B*a^2*b^3*d^4*e^2+10*B*
a*b^4*d^5*e+3*B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^10/(b*x+a)^5

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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)^11,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?

________________________________________________________________________________________

mupad [B]  time = 2.57, size = 1488, normalized size = 3.40

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)^11,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)/(6*e^7) - (d*((b^4*(A*b*e
+ 5*B*a*e - 4*B*b*d))/(6*e^6) - (B*b^5*d)/(6*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^
6) - (((A*b^5*e - 5*B*b^5*d + 5*B*a*b^4*e)/(5*e^7) - (B*b^5*d)/(5*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a +
b*x)*(d + e*x)^5) - (((A*a^5)/(10*e) - (d*((B*a^5 + 5*A*a^4*b)/(10*e) + (d*((d*((d*((d*((A*b^5 + 5*B*a*b^4)/(
10*e) - (B*b^5*d)/(10*e^2)))/e - (a*b^3*(A*b + 2*B*a))/(2*e)))/e + (a^2*b^2*(A*b + B*a))/e))/e - (a^3*b*(2*A*b
+ B*a))/(2*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((6*A*b^5*d^2*e - 10*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)/(7*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)/(7*e^7) - (d*((b
^4*(A*b*e + 5*B*a*e - 3*B*b*d))/(7*e^5) - (B*b^5*d)/(7*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*
x)*(d + e*x)^7) - (((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)/
(9*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)/(9*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)/(9*
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)/(9*e^7) - (d*((b
^4*(A*b*e + 5*B*a*e - B*b*d))/(9*e^3) - (B*b^5*d)/(9*e^3)))/e))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((
a + b*x)*(d + e*x)^9) - (((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)/(8*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)/(8*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)/(8*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 2*B*b*d))/(8*e^4) - (B*b^5*d)/(8*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*e^7*(a + b*x)*(d + e*x)^
4)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**11,x)

[Out]

Timed out

________________________________________________________________________________________