3.828 \(\int \frac{x^{7/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=306 \[ \frac{x^{9/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (A b-9 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{5/2} (A b-9 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 x^{3/2} (A b-9 a B)}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 \sqrt{x} (a+b x) (A b-9 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (a+b x) (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 \sqrt{a} b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(35*(A*b - 9*a*B)*x^(3/2))/(192*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*x^(9/2))/(4*a*b*(a + b*x)^
3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - 9*a*B)*x^(7/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) + (7*(A*b - 9*a*B)*x^(5/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(A*b - 9*a*B)*Sqrt[x]*(
a + b*x))/(64*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*(A*b - 9*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt
[a]])/(64*Sqrt[a]*b^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.150295, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {770, 78, 47, 50, 63, 205} \[ \frac{x^{9/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (A b-9 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{5/2} (A b-9 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 x^{3/2} (A b-9 a B)}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 \sqrt{x} (a+b x) (A b-9 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (a+b x) (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 \sqrt{a} b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(35*(A*b - 9*a*B)*x^(3/2))/(192*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*x^(9/2))/(4*a*b*(a + b*x)^
3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - 9*a*B)*x^(7/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) + (7*(A*b - 9*a*B)*x^(5/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(A*b - 9*a*B)*Sqrt[x]*(
a + b*x))/(64*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*(A*b - 9*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt
[a]])/(64*Sqrt[a]*b^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

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 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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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 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 )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{x^{7/2} (A+B x)}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (b^2 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{7/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (7 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{48 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{192 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{128 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (A b-9 a B) \sqrt{x} (a+b x)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{128 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (A b-9 a B) \sqrt{x} (a+b x)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 (A b-9 a B) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (A b-9 a B) x^{3/2}}{192 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) x^{9/2}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-9 a B) x^{7/2}}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (A b-9 a B) x^{5/2}}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (A b-9 a B) \sqrt{x} (a+b x)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (A b-9 a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 \sqrt{a} b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.034432, size = 79, normalized size = 0.26 \[ \frac{x^{9/2} \left (-9 a^4 (a B-A b)-(a+b x)^4 (A b-9 a B) \, _2F_1\left (4,\frac{9}{2};\frac{11}{2};-\frac{b x}{a}\right )\right )}{36 a^5 b (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x^(9/2)*(-9*a^4*(-(A*b) + a*B) - (A*b - 9*a*B)*(a + b*x)^4*Hypergeometric2F1[4, 9/2, 11/2, -((b*x)/a)]))/(36*
a^5*b*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.019, size = 368, normalized size = 1.2 \begin{align*} -{\frac{bx+a}{192\,{b}^{5}} \left ( 279\,A\sqrt{ab}{x}^{7/2}{b}^{4}-2511\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+511\,A\sqrt{ab}{x}^{5/2}a{b}^{3}-105\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}{b}^{5}-4599\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}-384\,B\sqrt{ab}{x}^{9/2}{b}^{4}+945\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}a{b}^{4}-420\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}a{b}^{4}+3780\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{3}+385\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}-630\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}-3465\,B\sqrt{ab}{x}^{3/2}{a}^{3}b+5670\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}-420\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}+3780\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{4}b+105\,A\sqrt{ab}\sqrt{x}{a}^{3}b-105\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{4}b-945\,B\sqrt{ab}\sqrt{x}{a}^{4}+945\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

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)^(5/2),x)

[Out]

-1/192*(279*A*(a*b)^(1/2)*x^(7/2)*b^4-2511*B*(a*b)^(1/2)*x^(7/2)*a*b^3+511*A*(a*b)^(1/2)*x^(5/2)*a*b^3-105*A*a
rctan(x^(1/2)*b/(a*b)^(1/2))*x^4*b^5-4599*B*(a*b)^(1/2)*x^(5/2)*a^2*b^2-384*B*(a*b)^(1/2)*x^(9/2)*b^4+945*B*ar
ctan(x^(1/2)*b/(a*b)^(1/2))*x^4*a*b^4-420*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a*b^4+3780*B*arctan(x^(1/2)*b/(a
*b)^(1/2))*x^3*a^2*b^3+385*A*(a*b)^(1/2)*x^(3/2)*a^2*b^2-630*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^2*b^3-3465*
B*(a*b)^(1/2)*x^(3/2)*a^3*b+5670*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^3*b^2-420*A*arctan(x^(1/2)*b/(a*b)^(1/2
))*x*a^3*b^2+3780*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^4*b+105*A*(a*b)^(1/2)*x^(1/2)*a^3*b-105*A*arctan(x^(1/2)
*b/(a*b)^(1/2))*a^4*b-945*B*(a*b)^(1/2)*x^(1/2)*a^4+945*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)*(b*x+a)/(a*b)^(1/
2)/b^5/((b*x+a)^2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50138, size = 1184, normalized size = 3.87 \begin{align*} \left [\frac{105 \,{\left (9 \, B a^{5} - A a^{4} b +{\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \,{\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \,{\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (9 \, B a^{4} b - A a^{3} 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 (384 \, B a b^{5} x^{4} + 945 \, B a^{5} b - 105 \, A a^{4} b^{2} + 279 \,{\left (9 \, B a^{2} b^{4} - A a b^{5}\right )} x^{3} + 511 \,{\left (9 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 385 \,{\left (9 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} \sqrt{x}}{384 \,{\left (a b^{10} x^{4} + 4 \, a^{2} b^{9} x^{3} + 6 \, a^{3} b^{8} x^{2} + 4 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}, \frac{105 \,{\left (9 \, B a^{5} - A a^{4} b +{\left (9 \, B a b^{4} - A b^{5}\right )} x^{4} + 4 \,{\left (9 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 6 \,{\left (9 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (9 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (384 \, B a b^{5} x^{4} + 945 \, B a^{5} b - 105 \, A a^{4} b^{2} + 279 \,{\left (9 \, B a^{2} b^{4} - A a b^{5}\right )} x^{3} + 511 \,{\left (9 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{2} + 385 \,{\left (9 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} \sqrt{x}}{192 \,{\left (a b^{10} x^{4} + 4 \, a^{2} b^{9} x^{3} + 6 \, a^{3} b^{8} x^{2} + 4 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}\right ] \end{align*}

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)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(105*(9*B*a^5 - A*a^4*b + (9*B*a*b^4 - A*b^5)*x^4 + 4*(9*B*a^2*b^3 - A*a*b^4)*x^3 + 6*(9*B*a^3*b^2 - A*
a^2*b^3)*x^2 + 4*(9*B*a^4*b - A*a^3*b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(38
4*B*a*b^5*x^4 + 945*B*a^5*b - 105*A*a^4*b^2 + 279*(9*B*a^2*b^4 - A*a*b^5)*x^3 + 511*(9*B*a^3*b^3 - A*a^2*b^4)*
x^2 + 385*(9*B*a^4*b^2 - A*a^3*b^3)*x)*sqrt(x))/(a*b^10*x^4 + 4*a^2*b^9*x^3 + 6*a^3*b^8*x^2 + 4*a^4*b^7*x + a^
5*b^6), 1/192*(105*(9*B*a^5 - A*a^4*b + (9*B*a*b^4 - A*b^5)*x^4 + 4*(9*B*a^2*b^3 - A*a*b^4)*x^3 + 6*(9*B*a^3*b
^2 - A*a^2*b^3)*x^2 + 4*(9*B*a^4*b - A*a^3*b^2)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (384*B*a*b^5*x^4
+ 945*B*a^5*b - 105*A*a^4*b^2 + 279*(9*B*a^2*b^4 - A*a*b^5)*x^3 + 511*(9*B*a^3*b^3 - A*a^2*b^4)*x^2 + 385*(9*B
*a^4*b^2 - A*a^3*b^3)*x)*sqrt(x))/(a*b^10*x^4 + 4*a^2*b^9*x^3 + 6*a^3*b^8*x^2 + 4*a^4*b^7*x + a^5*b^6)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.21753, size = 215, normalized size = 0.7 \begin{align*} \frac{2 \, B \sqrt{x}}{b^{5} \mathrm{sgn}\left (b x + a\right )} - \frac{35 \,{\left (9 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} b^{5} \mathrm{sgn}\left (b x + a\right )} + \frac{975 \, B a b^{3} x^{\frac{7}{2}} - 279 \, A b^{4} x^{\frac{7}{2}} + 2295 \, B a^{2} b^{2} x^{\frac{5}{2}} - 511 \, A a b^{3} x^{\frac{5}{2}} + 1929 \, B a^{3} b x^{\frac{3}{2}} - 385 \, A a^{2} b^{2} x^{\frac{3}{2}} + 561 \, B a^{4} \sqrt{x} - 105 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} b^{5} \mathrm{sgn}\left (b x + a\right )} \end{align*}

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)^(5/2),x, algorithm="giac")

[Out]

2*B*sqrt(x)/(b^5*sgn(b*x + a)) - 35/64*(9*B*a - A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5*sgn(b*x + a))
+ 1/192*(975*B*a*b^3*x^(7/2) - 279*A*b^4*x^(7/2) + 2295*B*a^2*b^2*x^(5/2) - 511*A*a*b^3*x^(5/2) + 1929*B*a^3*b
*x^(3/2) - 385*A*a^2*b^2*x^(3/2) + 561*B*a^4*sqrt(x) - 105*A*a^3*b*sqrt(x))/((b*x + a)^4*b^5*sgn(b*x + a))