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

[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))

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Rubi [A]  time = 0.0854049, antiderivative size = 190, normalized size of antiderivative = 1., 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 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 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 )^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*}

Mathematica [A]  time = 0.180712, size = 134, normalized size = 0.71 \[ \frac{(9 a B+A b) \left (105 (a+b x)^4 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )-\sqrt{a} \sqrt{b} \sqrt{x} \left (385 a^2 b x+105 a^3+511 a b^2 x^2+279 b^3 x^3\right )\right )}{1920 a^{3/2} b^{11/2} (a+b x)^4}+\frac{x^{9/2} (A b-a B)}{5 a b (a+b x)^5} \]

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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Maple [A]  time = 0.018, size = 150, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{ \left ( bx+a \right ) ^{5}} \left ({\frac{ \left ( 7\,Ab-193\,aB \right ){x}^{9/2}}{256\,ab}}-{\frac{ \left ( 79\,Ab+711\,aB \right ){x}^{7/2}}{384\,{b}^{2}}}-{\frac{7\,a \left ( Ab+9\,aB \right ){x}^{5/2}}{30\,{b}^{3}}}-{\frac{49\,{a}^{2} \left ( Ab+9\,aB \right ){x}^{3/2}}{384\,{b}^{4}}}-{\frac{ \left ( 7\,Ab+63\,aB \right ){a}^{3}\sqrt{x}}{256\,{b}^{5}}} \right ) }+{\frac{7\,A}{128\,a{b}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{63\,B}{128\,{b}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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)^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(x^(1/2)*b/
(a*b)^(1/2))*A+63/128/b^5/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76286, size = 1400, normalized size = 7.37 \begin{align*} \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 ] \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)^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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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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.16334, size = 209, normalized size = 1.1 \begin{align*} \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}} \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)^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)