3.775 \(\int \frac {x^{7/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=190 \[ \frac {7 (9 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{3/2} b^{11/2}}-\frac {7 \sqrt {x} (9 a B+A b)}{128 a b^5 (a+b x)}-\frac {7 x^{3/2} (9 a B+A b)}{192 a b^4 (a+b x)^2}-\frac {7 x^{5/2} (9 a B+A b)}{240 a b^3 (a+b x)^3}-\frac {x^{7/2} (9 a B+A b)}{40 a b^2 (a+b x)^4}+\frac {x^{9/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

1/5*(A*b-B*a)*x^(9/2)/a/b/(b*x+a)^5-1/40*(A*b+9*B*a)*x^(7/2)/a/b^2/(b*x+a)^4-7/240*(A*b+9*B*a)*x^(5/2)/a/b^3/(
b*x+a)^3-7/192*(A*b+9*B*a)*x^(3/2)/a/b^4/(b*x+a)^2+7/128*(A*b+9*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(3/2)/b
^(11/2)-7/128*(A*b+9*B*a)*x^(1/2)/a/b^5/(b*x+a)

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Rubi [A]  time = 0.09, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 78, 47, 63, 205} \[ \frac {7 (9 a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{3/2} b^{11/2}}-\frac {x^{7/2} (9 a B+A b)}{40 a b^2 (a+b x)^4}-\frac {7 x^{5/2} (9 a B+A b)}{240 a b^3 (a+b x)^3}-\frac {7 x^{3/2} (9 a B+A b)}{192 a b^4 (a+b x)^2}-\frac {7 \sqrt {x} (9 a B+A b)}{128 a b^5 (a+b x)}+\frac {x^{9/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((A*b - a*B)*x^(9/2))/(5*a*b*(a + b*x)^5) - ((A*b + 9*a*B)*x^(7/2))/(40*a*b^2*(a + b*x)^4) - (7*(A*b + 9*a*B)*
x^(5/2))/(240*a*b^3*(a + b*x)^3) - (7*(A*b + 9*a*B)*x^(3/2))/(192*a*b^4*(a + b*x)^2) - (7*(A*b + 9*a*B)*Sqrt[x
])/(128*a*b^5*(a + b*x)) + (7*(A*b + 9*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*a^(3/2)*b^(11/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin {align*} \int \frac {x^{7/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {x^{7/2} (A+B x)}{(a+b x)^6} \, dx\\ &=\frac {(A b-a B) x^{9/2}}{5 a b (a+b x)^5}+\frac {(A b+9 a B) \int \frac {x^{7/2}}{(a+b x)^5} \, dx}{10 a b}\\ &=\frac {(A b-a B) x^{9/2}}{5 a b (a+b x)^5}-\frac {(A b+9 a B) x^{7/2}}{40 a b^2 (a+b x)^4}+\frac {(7 (A b+9 a B)) \int \frac {x^{5/2}}{(a+b x)^4} \, dx}{80 a b^2}\\ &=\frac {(A b-a B) x^{9/2}}{5 a b (a+b x)^5}-\frac {(A b+9 a B) x^{7/2}}{40 a b^2 (a+b x)^4}-\frac {7 (A b+9 a B) x^{5/2}}{240 a b^3 (a+b x)^3}+\frac {(7 (A b+9 a B)) \int \frac {x^{3/2}}{(a+b x)^3} \, dx}{96 a b^3}\\ &=\frac {(A b-a B) x^{9/2}}{5 a b (a+b x)^5}-\frac {(A b+9 a B) x^{7/2}}{40 a b^2 (a+b x)^4}-\frac {7 (A b+9 a B) x^{5/2}}{240 a b^3 (a+b x)^3}-\frac {7 (A b+9 a B) x^{3/2}}{192 a b^4 (a+b x)^2}+\frac {(7 (A b+9 a B)) \int \frac {\sqrt {x}}{(a+b x)^2} \, dx}{128 a b^4}\\ &=\frac {(A b-a B) x^{9/2}}{5 a b (a+b x)^5}-\frac {(A b+9 a B) x^{7/2}}{40 a b^2 (a+b x)^4}-\frac {7 (A b+9 a B) x^{5/2}}{240 a b^3 (a+b x)^3}-\frac {7 (A b+9 a B) x^{3/2}}{192 a b^4 (a+b x)^2}-\frac {7 (A b+9 a B) \sqrt {x}}{128 a b^5 (a+b x)}+\frac {(7 (A b+9 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a b^5}\\ &=\frac {(A b-a B) x^{9/2}}{5 a b (a+b x)^5}-\frac {(A b+9 a B) x^{7/2}}{40 a b^2 (a+b x)^4}-\frac {7 (A b+9 a B) x^{5/2}}{240 a b^3 (a+b x)^3}-\frac {7 (A b+9 a B) x^{3/2}}{192 a b^4 (a+b x)^2}-\frac {7 (A b+9 a B) \sqrt {x}}{128 a b^5 (a+b x)}+\frac {(7 (A b+9 a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a b^5}\\ &=\frac {(A b-a B) x^{9/2}}{5 a b (a+b x)^5}-\frac {(A b+9 a B) x^{7/2}}{40 a b^2 (a+b x)^4}-\frac {7 (A b+9 a B) x^{5/2}}{240 a b^3 (a+b x)^3}-\frac {7 (A b+9 a B) x^{3/2}}{192 a b^4 (a+b x)^2}-\frac {7 (A b+9 a B) \sqrt {x}}{128 a b^5 (a+b x)}+\frac {7 (A b+9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{3/2} b^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 134, normalized size = 0.71 \[ \frac {(9 a B+A b) \left (105 (a+b x)^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )-\sqrt {a} \sqrt {b} \sqrt {x} \left (105 a^3+385 a^2 b x+511 a b^2 x^2+279 b^3 x^3\right )\right )}{1920 a^{3/2} b^{11/2} (a+b x)^4}+\frac {x^{9/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((A*b - a*B)*x^(9/2))/(5*a*b*(a + b*x)^5) + ((A*b + 9*a*B)*(-(Sqrt[a]*Sqrt[b]*Sqrt[x]*(105*a^3 + 385*a^2*b*x +
511*a*b^2*x^2 + 279*b^3*x^3)) + 105*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(1920*a^(3/2)*b^(11/2)*(a
+ b*x)^4)

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fricas [A]  time = 0.91, size = 639, normalized size = 3.36 \[ \left [-\frac {105 \, {\left (9 \, B a^{6} + A a^{5} b + {\left (9 \, B a b^{5} + A b^{6}\right )} x^{5} + 5 \, {\left (9 \, B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \, {\left (9 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (9 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (9 \, B a^{5} b + A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (945 \, B a^{6} b + 105 \, A a^{5} b^{2} + 15 \, {\left (193 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{4} + 790 \, {\left (9 \, B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{3} + 896 \, {\left (9 \, B a^{4} b^{3} + A a^{3} b^{4}\right )} x^{2} + 490 \, {\left (9 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{2} b^{11} x^{5} + 5 \, a^{3} b^{10} x^{4} + 10 \, a^{4} b^{9} x^{3} + 10 \, a^{5} b^{8} x^{2} + 5 \, a^{6} b^{7} x + a^{7} b^{6}\right )}}, -\frac {105 \, {\left (9 \, B a^{6} + A a^{5} b + {\left (9 \, B a b^{5} + A b^{6}\right )} x^{5} + 5 \, {\left (9 \, B a^{2} b^{4} + A a b^{5}\right )} x^{4} + 10 \, {\left (9 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (9 \, B a^{4} b^{2} + A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (9 \, B a^{5} b + A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (945 \, B a^{6} b + 105 \, A a^{5} b^{2} + 15 \, {\left (193 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{4} + 790 \, {\left (9 \, B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{3} + 896 \, {\left (9 \, B a^{4} b^{3} + A a^{3} b^{4}\right )} x^{2} + 490 \, {\left (9 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{2} b^{11} x^{5} + 5 \, a^{3} b^{10} x^{4} + 10 \, a^{4} b^{9} x^{3} + 10 \, a^{5} b^{8} x^{2} + 5 \, a^{6} b^{7} x + a^{7} b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(105*(9*B*a^6 + A*a^5*b + (9*B*a*b^5 + A*b^6)*x^5 + 5*(9*B*a^2*b^4 + A*a*b^5)*x^4 + 10*(9*B*a^3*b^3 +
A*a^2*b^4)*x^3 + 10*(9*B*a^4*b^2 + A*a^3*b^3)*x^2 + 5*(9*B*a^5*b + A*a^4*b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*
sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(945*B*a^6*b + 105*A*a^5*b^2 + 15*(193*B*a^2*b^5 - 7*A*a*b^6)*x^4 + 790*(9*
B*a^3*b^4 + A*a^2*b^5)*x^3 + 896*(9*B*a^4*b^3 + A*a^3*b^4)*x^2 + 490*(9*B*a^5*b^2 + A*a^4*b^3)*x)*sqrt(x))/(a^
2*b^11*x^5 + 5*a^3*b^10*x^4 + 10*a^4*b^9*x^3 + 10*a^5*b^8*x^2 + 5*a^6*b^7*x + a^7*b^6), -1/1920*(105*(9*B*a^6
+ A*a^5*b + (9*B*a*b^5 + A*b^6)*x^5 + 5*(9*B*a^2*b^4 + A*a*b^5)*x^4 + 10*(9*B*a^3*b^3 + A*a^2*b^4)*x^3 + 10*(9
*B*a^4*b^2 + A*a^3*b^3)*x^2 + 5*(9*B*a^5*b + A*a^4*b^2)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (945*B*a^
6*b + 105*A*a^5*b^2 + 15*(193*B*a^2*b^5 - 7*A*a*b^6)*x^4 + 790*(9*B*a^3*b^4 + A*a^2*b^5)*x^3 + 896*(9*B*a^4*b^
3 + A*a^3*b^4)*x^2 + 490*(9*B*a^5*b^2 + A*a^4*b^3)*x)*sqrt(x))/(a^2*b^11*x^5 + 5*a^3*b^10*x^4 + 10*a^4*b^9*x^3
+ 10*a^5*b^8*x^2 + 5*a^6*b^7*x + a^7*b^6)]

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giac [A]  time = 0.17, size = 155, normalized size = 0.82 \[ \frac {7 \, {\left (9 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a b^{5}} - \frac {2895 \, B a b^{4} x^{\frac {9}{2}} - 105 \, A b^{5} x^{\frac {9}{2}} + 7110 \, B a^{2} b^{3} x^{\frac {7}{2}} + 790 \, A a b^{4} x^{\frac {7}{2}} + 8064 \, B a^{3} b^{2} x^{\frac {5}{2}} + 896 \, A a^{2} b^{3} x^{\frac {5}{2}} + 4410 \, B a^{4} b x^{\frac {3}{2}} + 490 \, A a^{3} b^{2} x^{\frac {3}{2}} + 945 \, B a^{5} \sqrt {x} + 105 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

7/128*(9*B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a*b^5) - 1/1920*(2895*B*a*b^4*x^(9/2) - 105*A*b^5*x
^(9/2) + 7110*B*a^2*b^3*x^(7/2) + 790*A*a*b^4*x^(7/2) + 8064*B*a^3*b^2*x^(5/2) + 896*A*a^2*b^3*x^(5/2) + 4410*
B*a^4*b*x^(3/2) + 490*A*a^3*b^2*x^(3/2) + 945*B*a^5*sqrt(x) + 105*A*a^4*b*sqrt(x))/((b*x + a)^5*a*b^5)

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maple [A]  time = 0.06, size = 150, normalized size = 0.79 \[ \frac {7 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a \,b^{4}}+\frac {63 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, b^{5}}+\frac {\frac {\left (7 A b -193 B a \right ) x^{\frac {9}{2}}}{128 a b}-\frac {79 \left (A b +9 B a \right ) x^{\frac {7}{2}}}{192 b^{2}}-\frac {7 \left (A b +9 B a \right ) a \,x^{\frac {5}{2}}}{15 b^{3}}-\frac {49 \left (A b +9 B a \right ) a^{2} x^{\frac {3}{2}}}{192 b^{4}}-\frac {7 \left (A b +9 B a \right ) a^{3} \sqrt {x}}{128 b^{5}}}{\left (b x +a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*(1/256*(7*A*b-193*B*a)/a/b*x^(9/2)-79/384*(A*b+9*B*a)/b^2*x^(7/2)-7/30*a*(A*b+9*B*a)/b^3*x^(5/2)-49/384*a^2*
(A*b+9*B*a)/b^4*x^(3/2)-7/256*(A*b+9*B*a)*a^3/b^5*x^(1/2))/(b*x+a)^5+7/128/a/b^4/(a*b)^(1/2)*arctan(1/(a*b)^(1
/2)*b*x^(1/2))*A+63/128/b^5/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*B

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maxima [A]  time = 1.38, size = 198, normalized size = 1.04 \[ -\frac {15 \, {\left (193 \, B a b^{4} - 7 \, A b^{5}\right )} x^{\frac {9}{2}} + 790 \, {\left (9 \, B a^{2} b^{3} + A a b^{4}\right )} x^{\frac {7}{2}} + 896 \, {\left (9 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{\frac {5}{2}} + 490 \, {\left (9 \, B a^{4} b + A a^{3} b^{2}\right )} x^{\frac {3}{2}} + 105 \, {\left (9 \, B a^{5} + A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a b^{10} x^{5} + 5 \, a^{2} b^{9} x^{4} + 10 \, a^{3} b^{8} x^{3} + 10 \, a^{4} b^{7} x^{2} + 5 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} + \frac {7 \, {\left (9 \, B a + A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/1920*(15*(193*B*a*b^4 - 7*A*b^5)*x^(9/2) + 790*(9*B*a^2*b^3 + A*a*b^4)*x^(7/2) + 896*(9*B*a^3*b^2 + A*a^2*b
^3)*x^(5/2) + 490*(9*B*a^4*b + A*a^3*b^2)*x^(3/2) + 105*(9*B*a^5 + A*a^4*b)*sqrt(x))/(a*b^10*x^5 + 5*a^2*b^9*x
^4 + 10*a^3*b^8*x^3 + 10*a^4*b^7*x^2 + 5*a^5*b^6*x + a^6*b^5) + 7/128*(9*B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b)
)/(sqrt(a*b)*a*b^5)

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mupad [B]  time = 1.28, size = 173, normalized size = 0.91 \[ \frac {7\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b+9\,B\,a\right )}{128\,a^{3/2}\,b^{11/2}}-\frac {\frac {79\,x^{7/2}\,\left (A\,b+9\,B\,a\right )}{192\,b^2}+\frac {49\,a^2\,x^{3/2}\,\left (A\,b+9\,B\,a\right )}{192\,b^4}+\frac {7\,a^3\,\sqrt {x}\,\left (A\,b+9\,B\,a\right )}{128\,b^5}-\frac {x^{9/2}\,\left (7\,A\,b-193\,B\,a\right )}{128\,a\,b}+\frac {7\,a\,x^{5/2}\,\left (A\,b+9\,B\,a\right )}{15\,b^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(7*atan((b^(1/2)*x^(1/2))/a^(1/2))*(A*b + 9*B*a))/(128*a^(3/2)*b^(11/2)) - ((79*x^(7/2)*(A*b + 9*B*a))/(192*b^
2) + (49*a^2*x^(3/2)*(A*b + 9*B*a))/(192*b^4) + (7*a^3*x^(1/2)*(A*b + 9*B*a))/(128*b^5) - (x^(9/2)*(7*A*b - 19
3*B*a))/(128*a*b) + (7*a*x^(5/2)*(A*b + 9*B*a))/(15*b^3))/(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a
^2*b^3*x^3 + 5*a^4*b*x)

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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(x**(7/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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