3.16.33 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=193 \[ \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{28 e (d+e x)^7 (b d-a e)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

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Rubi [A]  time = 0.14, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {770, 78, 45, 37} \begin {gather*} \frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (-4 a B e+A b e+3 b B d)}{28 e (d+e x)^7 (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)}{8 e (d+e x)^8 (b d-a e)} \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)^9,x]

[Out]

-((B*d - A*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e*(b*d - a*e)*(d + e*x)^8) + ((3*b*B*d + A*b*e - 4
*a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(28*e*(b*d - a*e)^2*(d + e*x)^7) + (b*(3*b*B*d + A*b*e - 4*
a*B*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*e*(b*d - a*e)^3*(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)^9} \, 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)^9} \, 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}}{8 e (b d-a e) (d+e x)^8}+\frac {\left ((3 b B d+A b e-4 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}{4 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}}{8 e (b d-a e) (d+e x)^8}+\frac {(3 b B d+A b e-4 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 e (b d-a e)^2 (d+e x)^7}+\frac {\left ((3 b B d+A b e-4 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}{28 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}}{8 e (b d-a e) (d+e x)^8}+\frac {(3 b B d+A b e-4 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 e (b d-a e)^2 (d+e x)^7}+\frac {b (3 b B d+A b e-4 a B e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{168 e (b d-a e)^3 (d+e x)^6}\\ \end {align*}

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Mathematica [B]  time = 0.21, size = 466, normalized size = 2.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a^5 e^5 (7 A e+B (d+8 e x))+5 a^4 b e^4 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+2 a^3 b^2 e^3 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+6 a^2 b^3 e^2 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+a b^4 e \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 B \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )\right )}{168 e^7 (a+b x) (d+e x)^8} \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)^9,x]

[Out]

-1/168*(Sqrt[(a + b*x)^2]*(3*a^5*e^5*(7*A*e + B*(d + 8*e*x)) + 5*a^4*b*e^4*(3*A*e*(d + 8*e*x) + B*(d^2 + 8*d*e
*x + 28*e^2*x^2)) + 2*a^3*b^2*e^3*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 +
56*e^3*x^3)) + 6*a^2*b^3*e^2*(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x + 28*d^2*
e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + a*b^4*e*(3*A*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e
^4*x^4) + 5*B*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)) + b^5*(A*e*(d^5
+ 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + 3*B*(d^6 + 8*d^5*e*x + 28*d^4*e^
2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))))/(e^7*(a + b*x)*(d + e*x)^8)

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IntegrateAlgebraic [F]  time = 180.02, 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)^9,x]

[Out]

$Aborted

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fricas [B]  time = 0.43, size = 634, normalized size = 3.28 \begin {gather*} -\frac {84 \, B b^{5} e^{6} x^{6} + 3 \, B b^{5} d^{6} + 21 \, A a^{5} e^{6} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 3 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 56 \, {\left (3 \, B b^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 70 \, {\left (3 \, B b^{5} d^{2} e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 3 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 56 \, {\left (3 \, B b^{5} d^{3} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 3 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 28 \, {\left (3 \, B b^{5} d^{4} e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 3 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 8 \, {\left (3 \, B b^{5} d^{5} e + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 3 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{168 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} 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)^9,x, algorithm="fricas")

[Out]

-1/168*(84*B*b^5*e^6*x^6 + 3*B*b^5*d^6 + 21*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5*e + 3*(2*B*a^2*b^3 + A*a*b^4)*
d^4*e^2 + 6*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 3*(B*a^5 + 5*A*a^4*b)*d*e^5
+ 56*(3*B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 70*(3*B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*d*e^5 + 3*(2*B*
a^2*b^3 + A*a*b^4)*e^6)*x^4 + 56*(3*B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5)*d^2*e^4 + 3*(2*B*a^2*b^3 + A*a*b^4)*d*
e^5 + 6*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 28*(3*B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 3*(2*B*a^2*b^3
+ A*a*b^4)*d^2*e^4 + 6*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 5*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 8*(3*B*b^5*d^5*e +
(5*B*a*b^4 + A*b^5)*d^4*e^2 + 3*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 6*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 5*(B*a^
4*b + 2*A*a^3*b^2)*d*e^5 + 3*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e
^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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giac [B]  time = 0.22, size = 918, normalized size = 4.76

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)^9,x, algorithm="giac")

[Out]

-1/168*(84*B*b^5*x^6*e^6*sgn(b*x + a) + 168*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)
+ 168*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 84*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 24*B*b^5*d^5*x*e*sgn(b*x + a) + 3*B
*b^5*d^6*sgn(b*x + a) + 280*B*a*b^4*x^5*e^6*sgn(b*x + a) + 56*A*b^5*x^5*e^6*sgn(b*x + a) + 350*B*a*b^4*d*x^4*e
^5*sgn(b*x + a) + 70*A*b^5*d*x^4*e^5*sgn(b*x + a) + 280*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) + 56*A*b^5*d^2*x^3*e^
4*sgn(b*x + a) + 140*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 28*A*b^5*d^3*x^2*e^3*sgn(b*x + a) + 40*B*a*b^4*d^4*x*e
^2*sgn(b*x + a) + 8*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) + 4
20*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 210*A*a*b^4*x^4*e^6*sgn(b*x + a) + 336*B*a^2*b^3*d*x^3*e^5*sgn(b*x + a) +
168*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 168*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x + a) + 84*A*a*b^4*d^2*x^2*e^4*sgn(b*x +
a) + 48*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 24*A*a*b^4*d^3*x*e^3*sgn(b*x + a) + 6*B*a^2*b^3*d^4*e^2*sgn(b*x +
a) + 3*A*a*b^4*d^4*e^2*sgn(b*x + a) + 336*B*a^3*b^2*x^3*e^6*sgn(b*x + a) + 336*A*a^2*b^3*x^3*e^6*sgn(b*x + a)
+ 168*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 168*A*a^2*b^3*d*x^2*e^5*sgn(b*x + a) + 48*B*a^3*b^2*d^2*x*e^4*sgn(b*x
+ a) + 48*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 6*B*a^3*b^2*d^3*e^3*sgn(b*x + a) + 6*A*a^2*b^3*d^3*e^3*sgn(b*x +
a) + 140*B*a^4*b*x^2*e^6*sgn(b*x + a) + 280*A*a^3*b^2*x^2*e^6*sgn(b*x + a) + 40*B*a^4*b*d*x*e^5*sgn(b*x + a)
+ 80*A*a^3*b^2*d*x*e^5*sgn(b*x + a) + 5*B*a^4*b*d^2*e^4*sgn(b*x + a) + 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) + 24*
B*a^5*x*e^6*sgn(b*x + a) + 120*A*a^4*b*x*e^6*sgn(b*x + a) + 3*B*a^5*d*e^5*sgn(b*x + a) + 15*A*a^4*b*d*e^5*sgn(
b*x + a) + 21*A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^8

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maple [B]  time = 0.05, size = 688, normalized size = 3.56 \begin {gather*} -\frac {\left (84 B \,b^{5} e^{6} x^{6}+56 A \,b^{5} e^{6} x^{5}+280 B a \,b^{4} e^{6} x^{5}+168 B \,b^{5} d \,e^{5} x^{5}+210 A a \,b^{4} e^{6} x^{4}+70 A \,b^{5} d \,e^{5} x^{4}+420 B \,a^{2} b^{3} e^{6} x^{4}+350 B a \,b^{4} d \,e^{5} x^{4}+210 B \,b^{5} d^{2} e^{4} x^{4}+336 A \,a^{2} b^{3} e^{6} x^{3}+168 A a \,b^{4} d \,e^{5} x^{3}+56 A \,b^{5} d^{2} e^{4} x^{3}+336 B \,a^{3} b^{2} e^{6} x^{3}+336 B \,a^{2} b^{3} d \,e^{5} x^{3}+280 B a \,b^{4} d^{2} e^{4} x^{3}+168 B \,b^{5} d^{3} e^{3} x^{3}+280 A \,a^{3} b^{2} e^{6} x^{2}+168 A \,a^{2} b^{3} d \,e^{5} x^{2}+84 A a \,b^{4} d^{2} e^{4} x^{2}+28 A \,b^{5} d^{3} e^{3} x^{2}+140 B \,a^{4} b \,e^{6} x^{2}+168 B \,a^{3} b^{2} d \,e^{5} x^{2}+168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+140 B a \,b^{4} d^{3} e^{3} x^{2}+84 B \,b^{5} d^{4} e^{2} x^{2}+120 A \,a^{4} b \,e^{6} x +80 A \,a^{3} b^{2} d \,e^{5} x +48 A \,a^{2} b^{3} d^{2} e^{4} x +24 A a \,b^{4} d^{3} e^{3} x +8 A \,b^{5} d^{4} e^{2} x +24 B \,a^{5} e^{6} x +40 B \,a^{4} b d \,e^{5} x +48 B \,a^{3} b^{2} d^{2} e^{4} x +48 B \,a^{2} b^{3} d^{3} e^{3} x +40 B a \,b^{4} d^{4} e^{2} x +24 B \,b^{5} d^{5} e x +21 A \,a^{5} e^{6}+15 A \,a^{4} b d \,e^{5}+10 A \,a^{3} b^{2} d^{2} e^{4}+6 A \,a^{2} b^{3} d^{3} e^{3}+3 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +3 B \,a^{5} d \,e^{5}+5 B \,a^{4} b \,d^{2} e^{4}+6 B \,a^{3} b^{2} d^{3} e^{3}+6 B \,a^{2} b^{3} d^{4} e^{2}+5 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}}}{168 \left (e x +d \right )^{8} \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)^9,x)

[Out]

-1/168/e^7*(84*B*b^5*e^6*x^6+56*A*b^5*e^6*x^5+280*B*a*b^4*e^6*x^5+168*B*b^5*d*e^5*x^5+210*A*a*b^4*e^6*x^4+70*A
*b^5*d*e^5*x^4+420*B*a^2*b^3*e^6*x^4+350*B*a*b^4*d*e^5*x^4+210*B*b^5*d^2*e^4*x^4+336*A*a^2*b^3*e^6*x^3+168*A*a
*b^4*d*e^5*x^3+56*A*b^5*d^2*e^4*x^3+336*B*a^3*b^2*e^6*x^3+336*B*a^2*b^3*d*e^5*x^3+280*B*a*b^4*d^2*e^4*x^3+168*
B*b^5*d^3*e^3*x^3+280*A*a^3*b^2*e^6*x^2+168*A*a^2*b^3*d*e^5*x^2+84*A*a*b^4*d^2*e^4*x^2+28*A*b^5*d^3*e^3*x^2+14
0*B*a^4*b*e^6*x^2+168*B*a^3*b^2*d*e^5*x^2+168*B*a^2*b^3*d^2*e^4*x^2+140*B*a*b^4*d^3*e^3*x^2+84*B*b^5*d^4*e^2*x
^2+120*A*a^4*b*e^6*x+80*A*a^3*b^2*d*e^5*x+48*A*a^2*b^3*d^2*e^4*x+24*A*a*b^4*d^3*e^3*x+8*A*b^5*d^4*e^2*x+24*B*a
^5*e^6*x+40*B*a^4*b*d*e^5*x+48*B*a^3*b^2*d^2*e^4*x+48*B*a^2*b^3*d^3*e^3*x+40*B*a*b^4*d^4*e^2*x+24*B*b^5*d^5*e*
x+21*A*a^5*e^6+15*A*a^4*b*d*e^5+10*A*a^3*b^2*d^2*e^4+6*A*a^2*b^3*d^3*e^3+3*A*a*b^4*d^4*e^2+A*b^5*d^5*e+3*B*a^5
*d*e^5+5*B*a^4*b*d^2*e^4+6*B*a^3*b^2*d^3*e^3+6*B*a^2*b^3*d^4*e^2+5*B*a*b^4*d^5*e+3*B*b^5*d^6)*((b*x+a)^2)^(5/2
)/(e*x+d)^8/(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)^9,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 = 2.54, size = 1489, normalized size = 7.72

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

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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)**9,x)

[Out]

Timed out

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