3.16.32 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=106 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 (d+e x)^6 (b d-a e)^2}+\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2} (B d-A e)}{7 (d+e x)^7 (b d-a e)^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {769, 646, 37} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B)}{6 (d+e x)^6 (b d-a e)^2}+\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2} (B d-A e)}{7 (d+e x)^7 (b d-a e)^2} \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)^8,x]

[Out]

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

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 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 769

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(-2*c*(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)^2), x] + Dist[(2*c*f -
b*g)/(2*c*d - b*e), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x]
&& EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 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)^8} \, dx &=\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e)^2 (d+e x)^7}+\frac {(A b-a B) \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx}{b d-a e}\\ &=\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e)^2 (d+e x)^7}+\frac {\left ((A b-a B) \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}{b^4 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (b d-a e)^2 (d+e x)^6}+\frac {(B d-A e) \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e)^2 (d+e x)^7}\\ \end {align*}

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Mathematica [B]  time = 0.20, size = 465, normalized size = 4.39 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5 e^5 (6 A e+B (d+7 e x))+a^4 b e^4 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+a^3 b^2 e^3 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+a^2 b^3 e^2 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+a b^4 e \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+6 B \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )\right )}{42 e^7 (a+b x) (d+e x)^7} \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)^8,x]

[Out]

-1/42*(Sqrt[(a + b*x)^2]*(a^5*e^5*(6*A*e + B*(d + 7*e*x)) + a^4*b*e^4*(5*A*e*(d + 7*e*x) + 2*B*(d^2 + 7*d*e*x
+ 21*e^2*x^2)) + a^3*b^2*e^3*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^
3*x^3)) + a^2*b^3*e^2*(3*A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^2*e^2
*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + a*b^4*e*(2*A*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*
x^4) + 5*B*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + b^5*(A*e*(d^5 +
7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 6*B*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x
^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6))))/(e^7*(a + b*x)*(d + e*x)^7)

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IntegrateAlgebraic [F]  time = 180.04, 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)^8,x]

[Out]

$Aborted

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fricas [B]  time = 0.43, size = 621, normalized size = 5.86 \begin {gather*} -\frac {42 \, B b^{5} e^{6} x^{6} + 6 \, B b^{5} d^{6} + 6 \, A a^{5} e^{6} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 3 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 2 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 21 \, {\left (6 \, B b^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 35 \, {\left (6 \, B b^{5} d^{2} e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 35 \, {\left (6 \, B b^{5} d^{3} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 3 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 21 \, {\left (6 \, B b^{5} d^{4} e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 3 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 2 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 7 \, {\left (6 \, B b^{5} d^{5} e + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 3 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 2 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{42 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} 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)^8,x, algorithm="fricas")

[Out]

-1/42*(42*B*b^5*e^6*x^6 + 6*B*b^5*d^6 + 6*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5*e + 2*(2*B*a^2*b^3 + A*a*b^4)*d^
4*e^2 + 3*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 2*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + (B*a^5 + 5*A*a^4*b)*d*e^5 + 21
*(6*B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 35*(6*B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*d*e^5 + 2*(2*B*a^2*
b^3 + A*a*b^4)*e^6)*x^4 + 35*(6*B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5)*d^2*e^4 + 2*(2*B*a^2*b^3 + A*a*b^4)*d*e^5
+ 3*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 21*(6*B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 2*(2*B*a^2*b^3 + A*
a*b^4)*d^2*e^4 + 3*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 2*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 7*(6*B*b^5*d^5*e + (5*
B*a*b^4 + A*b^5)*d^4*e^2 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 3*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 2*(B*a^4*b
+ 2*A*a^3*b^2)*d*e^5 + (B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^
4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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giac [B]  time = 0.22, size = 917, normalized size = 8.65

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)^8,x, algorithm="giac")

[Out]

-1/42*(42*B*b^5*x^6*e^6*sgn(b*x + a) + 126*B*b^5*d*x^5*e^5*sgn(b*x + a) + 210*B*b^5*d^2*x^4*e^4*sgn(b*x + a) +
210*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 126*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 42*B*b^5*d^5*x*e*sgn(b*x + a) + 6*B
*b^5*d^6*sgn(b*x + a) + 105*B*a*b^4*x^5*e^6*sgn(b*x + a) + 21*A*b^5*x^5*e^6*sgn(b*x + a) + 175*B*a*b^4*d*x^4*e
^5*sgn(b*x + a) + 35*A*b^5*d*x^4*e^5*sgn(b*x + a) + 175*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) + 35*A*b^5*d^2*x^3*e^
4*sgn(b*x + a) + 105*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 21*A*b^5*d^3*x^2*e^3*sgn(b*x + a) + 35*B*a*b^4*d^4*x*e
^2*sgn(b*x + a) + 7*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) + 1
40*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 70*A*a*b^4*x^4*e^6*sgn(b*x + a) + 140*B*a^2*b^3*d*x^3*e^5*sgn(b*x + a) + 7
0*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 84*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x + a) + 42*A*a*b^4*d^2*x^2*e^4*sgn(b*x + a)
+ 28*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 14*A*a*b^4*d^3*x*e^3*sgn(b*x + a) + 4*B*a^2*b^3*d^4*e^2*sgn(b*x + a)
+ 2*A*a*b^4*d^4*e^2*sgn(b*x + a) + 105*B*a^3*b^2*x^3*e^6*sgn(b*x + a) + 105*A*a^2*b^3*x^3*e^6*sgn(b*x + a) + 6
3*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 63*A*a^2*b^3*d*x^2*e^5*sgn(b*x + a) + 21*B*a^3*b^2*d^2*x*e^4*sgn(b*x + a)
+ 21*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 3*B*a^3*b^2*d^3*e^3*sgn(b*x + a) + 3*A*a^2*b^3*d^3*e^3*sgn(b*x + a) +
42*B*a^4*b*x^2*e^6*sgn(b*x + a) + 84*A*a^3*b^2*x^2*e^6*sgn(b*x + a) + 14*B*a^4*b*d*x*e^5*sgn(b*x + a) + 28*A*
a^3*b^2*d*x*e^5*sgn(b*x + a) + 2*B*a^4*b*d^2*e^4*sgn(b*x + a) + 4*A*a^3*b^2*d^2*e^4*sgn(b*x + a) + 7*B*a^5*x*e
^6*sgn(b*x + a) + 35*A*a^4*b*x*e^6*sgn(b*x + a) + B*a^5*d*e^5*sgn(b*x + a) + 5*A*a^4*b*d*e^5*sgn(b*x + a) + 6*
A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^7

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maple [B]  time = 0.06, size = 687, normalized size = 6.48 \begin {gather*} -\frac {\left (42 B \,b^{5} e^{6} x^{6}+21 A \,b^{5} e^{6} x^{5}+105 B a \,b^{4} e^{6} x^{5}+126 B \,b^{5} d \,e^{5} x^{5}+70 A a \,b^{4} e^{6} x^{4}+35 A \,b^{5} d \,e^{5} x^{4}+140 B \,a^{2} b^{3} e^{6} x^{4}+175 B a \,b^{4} d \,e^{5} x^{4}+210 B \,b^{5} d^{2} e^{4} x^{4}+105 A \,a^{2} b^{3} e^{6} x^{3}+70 A a \,b^{4} d \,e^{5} x^{3}+35 A \,b^{5} d^{2} e^{4} x^{3}+105 B \,a^{3} b^{2} e^{6} x^{3}+140 B \,a^{2} b^{3} d \,e^{5} x^{3}+175 B a \,b^{4} d^{2} e^{4} x^{3}+210 B \,b^{5} d^{3} e^{3} x^{3}+84 A \,a^{3} b^{2} e^{6} x^{2}+63 A \,a^{2} b^{3} d \,e^{5} x^{2}+42 A a \,b^{4} d^{2} e^{4} x^{2}+21 A \,b^{5} d^{3} e^{3} x^{2}+42 B \,a^{4} b \,e^{6} x^{2}+63 B \,a^{3} b^{2} d \,e^{5} x^{2}+84 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+105 B a \,b^{4} d^{3} e^{3} x^{2}+126 B \,b^{5} d^{4} e^{2} x^{2}+35 A \,a^{4} b \,e^{6} x +28 A \,a^{3} b^{2} d \,e^{5} x +21 A \,a^{2} b^{3} d^{2} e^{4} x +14 A a \,b^{4} d^{3} e^{3} x +7 A \,b^{5} d^{4} e^{2} x +7 B \,a^{5} e^{6} x +14 B \,a^{4} b d \,e^{5} x +21 B \,a^{3} b^{2} d^{2} e^{4} x +28 B \,a^{2} b^{3} d^{3} e^{3} x +35 B a \,b^{4} d^{4} e^{2} x +42 B \,b^{5} d^{5} e x +6 A \,a^{5} e^{6}+5 A \,a^{4} b d \,e^{5}+4 A \,a^{3} b^{2} d^{2} e^{4}+3 A \,a^{2} b^{3} d^{3} e^{3}+2 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +B \,a^{5} d \,e^{5}+2 B \,a^{4} b \,d^{2} e^{4}+3 B \,a^{3} b^{2} d^{3} e^{3}+4 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +6 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 \left (e x +d \right )^{7} \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)^8,x)

[Out]

-1/42*(42*B*b^5*e^6*x^6+21*A*b^5*e^6*x^5+105*B*a*b^4*e^6*x^5+126*B*b^5*d*e^5*x^5+70*A*a*b^4*e^6*x^4+35*A*b^5*d
*e^5*x^4+140*B*a^2*b^3*e^6*x^4+175*B*a*b^4*d*e^5*x^4+210*B*b^5*d^2*e^4*x^4+105*A*a^2*b^3*e^6*x^3+70*A*a*b^4*d*
e^5*x^3+35*A*b^5*d^2*e^4*x^3+105*B*a^3*b^2*e^6*x^3+140*B*a^2*b^3*d*e^5*x^3+175*B*a*b^4*d^2*e^4*x^3+210*B*b^5*d
^3*e^3*x^3+84*A*a^3*b^2*e^6*x^2+63*A*a^2*b^3*d*e^5*x^2+42*A*a*b^4*d^2*e^4*x^2+21*A*b^5*d^3*e^3*x^2+42*B*a^4*b*
e^6*x^2+63*B*a^3*b^2*d*e^5*x^2+84*B*a^2*b^3*d^2*e^4*x^2+105*B*a*b^4*d^3*e^3*x^2+126*B*b^5*d^4*e^2*x^2+35*A*a^4
*b*e^6*x+28*A*a^3*b^2*d*e^5*x+21*A*a^2*b^3*d^2*e^4*x+14*A*a*b^4*d^3*e^3*x+7*A*b^5*d^4*e^2*x+7*B*a^5*e^6*x+14*B
*a^4*b*d*e^5*x+21*B*a^3*b^2*d^2*e^4*x+28*B*a^2*b^3*d^3*e^3*x+35*B*a*b^4*d^4*e^2*x+42*B*b^5*d^5*e*x+6*A*a^5*e^6
+5*A*a^4*b*d*e^5+4*A*a^3*b^2*d^2*e^4+3*A*a^2*b^3*d^3*e^3+2*A*a*b^4*d^4*e^2+A*b^5*d^5*e+B*a^5*d*e^5+2*B*a^4*b*d
^2*e^4+3*B*a^3*b^2*d^3*e^3+4*B*a^2*b^3*d^4*e^2+5*B*a*b^4*d^5*e+6*B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^7/e^7/(b
*x+a)^5

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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)^8,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 = 2.59, size = 1489, normalized size = 14.05

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

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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)**8,x)

[Out]

Timed out

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