3.8.71 \(\int \frac {A+B x}{x^{3/2} (a^2+2 a b x+b^2 x^2)^2} \, dx\) [771]

Optimal. Leaf size=157 \[ -\frac {5 (7 A b-a B)}{8 a^4 b \sqrt {x}}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {5 (7 A b-a B)}{24 a^3 b \sqrt {x} (a+b x)}-\frac {5 (7 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{9/2} \sqrt {b}} \]

[Out]

-5/8*(7*A*b-B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(9/2)/b^(1/2)-5/8*(7*A*b-B*a)/a^4/b/x^(1/2)+1/3*(A*b-B*a)/a
/b/(b*x+a)^3/x^(1/2)+1/12*(7*A*b-B*a)/a^2/b/(b*x+a)^2/x^(1/2)+5/24*(7*A*b-B*a)/a^3/b/(b*x+a)/x^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 79, 44, 53, 65, 211} \begin {gather*} -\frac {5 (7 A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{9/2} \sqrt {b}}-\frac {5 (7 A b-a B)}{8 a^4 b \sqrt {x}}+\frac {5 (7 A b-a B)}{24 a^3 b \sqrt {x} (a+b x)}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(7*A*b - a*B))/(8*a^4*b*Sqrt[x]) + (A*b - a*B)/(3*a*b*Sqrt[x]*(a + b*x)^3) + (7*A*b - a*B)/(12*a^2*b*Sqrt[
x]*(a + b*x)^2) + (5*(7*A*b - a*B))/(24*a^3*b*Sqrt[x]*(a + b*x)) - (5*(7*A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/S
qrt[a]])/(8*a^(9/2)*Sqrt[b])

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 44

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*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 79

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] || L
tQ[p, n]))))

Rule 211

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

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {A+B x}{x^{3/2} (a+b x)^4} \, dx\\ &=\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}-\frac {\left (-\frac {7 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} (a+b x)^3} \, dx}{3 a b}\\ &=\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {(5 (7 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)^2} \, dx}{24 a^2 b}\\ &=\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {5 (7 A b-a B)}{24 a^3 b \sqrt {x} (a+b x)}+\frac {(5 (7 A b-a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{16 a^3 b}\\ &=-\frac {5 (7 A b-a B)}{8 a^4 b \sqrt {x}}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {5 (7 A b-a B)}{24 a^3 b \sqrt {x} (a+b x)}-\frac {(5 (7 A b-a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{16 a^4}\\ &=-\frac {5 (7 A b-a B)}{8 a^4 b \sqrt {x}}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {5 (7 A b-a B)}{24 a^3 b \sqrt {x} (a+b x)}-\frac {(5 (7 A b-a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{8 a^4}\\ &=-\frac {5 (7 A b-a B)}{8 a^4 b \sqrt {x}}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {5 (7 A b-a B)}{24 a^3 b \sqrt {x} (a+b x)}-\frac {5 (7 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{9/2} \sqrt {b}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 112, normalized size = 0.71 \begin {gather*} \frac {-105 A b^3 x^3+5 a b^2 x^2 (-56 A+3 B x)+a^3 (-48 A+33 B x)+a^2 b x (-231 A+40 B x)}{24 a^4 \sqrt {x} (a+b x)^3}+\frac {5 (-7 A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{9/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-105*A*b^3*x^3 + 5*a*b^2*x^2*(-56*A + 3*B*x) + a^3*(-48*A + 33*B*x) + a^2*b*x*(-231*A + 40*B*x))/(24*a^4*Sqrt
[x]*(a + b*x)^3) + (5*(-7*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(9/2)*Sqrt[b])

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Maple [A]
time = 0.68, size = 105, normalized size = 0.67

method	result	size
derivativedivides	\(-\frac {2 A}{a^{4} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {19}{16} A \,b^{3}-\frac {5}{16} B a \,b^{2}\right ) x^{\frac {5}{2}}+\frac {a b \left (17 A b -5 B a \right ) x^{\frac {3}{2}}}{6}+\left (\frac {29}{16} A \,a^{2} b -\frac {11}{16} B \,a^{3}\right ) \sqrt {x}}{\left (b x +a \right )^{3}}+\frac {5 \left (7 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{16 \sqrt {a b}}\right )}{a^{4}}\)	\(105\)
default	\(-\frac {2 A}{a^{4} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {19}{16} A \,b^{3}-\frac {5}{16} B a \,b^{2}\right ) x^{\frac {5}{2}}+\frac {a b \left (17 A b -5 B a \right ) x^{\frac {3}{2}}}{6}+\left (\frac {29}{16} A \,a^{2} b -\frac {11}{16} B \,a^{3}\right ) \sqrt {x}}{\left (b x +a \right )^{3}}+\frac {5 \left (7 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{16 \sqrt {a b}}\right )}{a^{4}}\)	\(105\)
risch	\(-\frac {2 A}{a^{4} \sqrt {x}}-\frac {19 x^{\frac {5}{2}} A \,b^{3}}{8 a^{4} \left (b x +a \right )^{3}}+\frac {5 x^{\frac {5}{2}} B \,b^{2}}{8 a^{3} \left (b x +a \right )^{3}}-\frac {17 A \,x^{\frac {3}{2}} b^{2}}{3 a^{3} \left (b x +a \right )^{3}}+\frac {5 B \,x^{\frac {3}{2}} b}{3 a^{2} \left (b x +a \right )^{3}}-\frac {29 \sqrt {x}\, A b}{8 a^{2} \left (b x +a \right )^{3}}+\frac {11 \sqrt {x}\, B}{8 a \left (b x +a \right )^{3}}-\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A b}{8 a^{4} \sqrt {a b}}+\frac {5 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B}{8 a^{3} \sqrt {a b}}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

-2*A/a^4/x^(1/2)-2/a^4*(((19/16*A*b^3-5/16*B*a*b^2)*x^(5/2)+1/6*a*b*(17*A*b-5*B*a)*x^(3/2)+(29/16*A*a^2*b-11/1
6*B*a^3)*x^(1/2))/(b*x+a)^3+5/16*(7*A*b-B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))

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Maxima [A]
time = 0.49, size = 132, normalized size = 0.84 \begin {gather*} -\frac {48 \, A a^{3} - 15 \, {\left (B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 40 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{2} - 33 \, {\left (B a^{3} - 7 \, A a^{2} b\right )} x}{24 \, {\left (a^{4} b^{3} x^{\frac {7}{2}} + 3 \, a^{5} b^{2} x^{\frac {5}{2}} + 3 \, a^{6} b x^{\frac {3}{2}} + a^{7} \sqrt {x}\right )}} + \frac {5 \, {\left (B a - 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(48*A*a^3 - 15*(B*a*b^2 - 7*A*b^3)*x^3 - 40*(B*a^2*b - 7*A*a*b^2)*x^2 - 33*(B*a^3 - 7*A*a^2*b)*x)/(a^4*b
^3*x^(7/2) + 3*a^5*b^2*x^(5/2) + 3*a^6*b*x^(3/2) + a^7*sqrt(x)) + 5/8*(B*a - 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b)
)/(sqrt(a*b)*a^4)

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Fricas [A]
time = 2.98, size = 445, normalized size = 2.83 \begin {gather*} \left [\frac {15 \, {\left ({\left (B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 3 \, {\left (B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 3 \, {\left (B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + {\left (B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (48 \, A a^{4} b - 15 \, {\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{3} - 40 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} - 33 \, {\left (B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {x}}{48 \, {\left (a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{3} + 3 \, a^{7} b^{2} x^{2} + a^{8} b x\right )}}, -\frac {15 \, {\left ({\left (B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 3 \, {\left (B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 3 \, {\left (B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + {\left (B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (48 \, A a^{4} b - 15 \, {\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{3} - 40 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} - 33 \, {\left (B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{3} + 3 \, a^{7} b^{2} x^{2} + a^{8} b x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*((B*a*b^3 - 7*A*b^4)*x^4 + 3*(B*a^2*b^2 - 7*A*a*b^3)*x^3 + 3*(B*a^3*b - 7*A*a^2*b^2)*x^2 + (B*a^4 -
7*A*a^3*b)*x)*sqrt(-a*b)*log((b*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(48*A*a^4*b - 15*(B*a^2*b^3 - 7*A
*a*b^4)*x^3 - 40*(B*a^3*b^2 - 7*A*a^2*b^3)*x^2 - 33*(B*a^4*b - 7*A*a^3*b^2)*x)*sqrt(x))/(a^5*b^4*x^4 + 3*a^6*b
^3*x^3 + 3*a^7*b^2*x^2 + a^8*b*x), -1/24*(15*((B*a*b^3 - 7*A*b^4)*x^4 + 3*(B*a^2*b^2 - 7*A*a*b^3)*x^3 + 3*(B*a
^3*b - 7*A*a^2*b^2)*x^2 + (B*a^4 - 7*A*a^3*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (48*A*a^4*b - 15*(B
*a^2*b^3 - 7*A*a*b^4)*x^3 - 40*(B*a^3*b^2 - 7*A*a^2*b^3)*x^2 - 33*(B*a^4*b - 7*A*a^3*b^2)*x)*sqrt(x))/(a^5*b^4
*x^4 + 3*a^6*b^3*x^3 + 3*a^7*b^2*x^2 + a^8*b*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2660 vs. \(2 (146) = 292\).
time = 93.63, size = 2660, normalized size = 16.94 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/sqrt(x) + 2*B*sqrt(x))/a**
4, Eq(b, 0)), ((-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2)))/b**4, Eq(a, 0)), (-105*A*a**3*b*sqrt(x)*log(sqrt(x) - sq
rt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b
) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) + 105*A*a**3*b*sqrt(x)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt
(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a
/b)) - 96*A*a**3*b*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**
3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 315*A*a**2*b**2*x**(3/2)*log(sqrt(x) - sqrt(-a/b))
/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a*
*4*b**4*x**(7/2)*sqrt(-a/b)) + 315*A*a**2*b**2*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b
) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b))
- 462*A*a**2*b**2*x*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b*
*3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 315*A*a*b**3*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(
48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4
*b**4*x**(7/2)*sqrt(-a/b)) + 315*A*a*b**3*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 1
44*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 560
*A*a*b**3*x**2*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x*
*(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 105*A*b**4*x**(7/2)*log(sqrt(x) - sqrt(-a/b))/(48*a**7
*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x
**(7/2)*sqrt(-a/b)) + 105*A*b**4*x**(7/2)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b
**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 210*A*b**4*x
**3*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt
(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) + 15*B*a**4*sqrt(x)*log(sqrt(x) - sqrt(-a/b))/(48*a**7*b*sqrt(x)*sq
rt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(
-a/b)) - 15*B*a**4*sqrt(x)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sq
rt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) + 45*B*a**3*b*x**(3/2)*log(sq
rt(x) - sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)
*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 45*B*a**3*b*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sq
rt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/
2)*sqrt(-a/b)) + 66*B*a**3*b*x*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) +
144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) + 45*B*a**2*b**2*x**(5/2)*log(sqrt(x) -
sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a
/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 45*B*a**2*b**2*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)
*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sq
rt(-a/b)) + 80*B*a**2*b**2*x**2*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) +
144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) + 15*B*a*b**3*x**(7/2)*log(sqrt(x) - sq
rt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b
) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 15*B*a*b**3*x**(7/2)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt
(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a
/b)) + 30*B*a*b**3*x**3*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**
5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)), True))

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Giac [A]
time = 0.79, size = 110, normalized size = 0.70 \begin {gather*} \frac {5 \, {\left (B a - 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} - \frac {2 \, A}{a^{4} \sqrt {x}} + \frac {15 \, B a b^{2} x^{\frac {5}{2}} - 57 \, A b^{3} x^{\frac {5}{2}} + 40 \, B a^{2} b x^{\frac {3}{2}} - 136 \, A a b^{2} x^{\frac {3}{2}} + 33 \, B a^{3} \sqrt {x} - 87 \, A a^{2} b \sqrt {x}}{24 \, {\left (b x + a\right )}^{3} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8*(B*a - 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 2*A/(a^4*sqrt(x)) + 1/24*(15*B*a*b^2*x^(5/2) -
57*A*b^3*x^(5/2) + 40*B*a^2*b*x^(3/2) - 136*A*a*b^2*x^(3/2) + 33*B*a^3*sqrt(x) - 87*A*a^2*b*sqrt(x))/((b*x +
a)^3*a^4)

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Mupad [B]
time = 1.28, size = 147, normalized size = 0.94 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {11\,x\,\left (7\,A\,b-B\,a\right )}{8\,a^2}+\frac {5\,b^2\,x^3\,\left (7\,A\,b-B\,a\right )}{8\,a^4}+\frac {5\,b\,x^2\,\left (7\,A\,b-B\,a\right )}{3\,a^3}}{a^3\,\sqrt {x}+b^3\,x^{7/2}+3\,a^2\,b\,x^{3/2}+3\,a\,b^2\,x^{5/2}}-\frac {5\,\mathrm {atan}\left (\frac {5\,\sqrt {b}\,\sqrt {x}\,\left (7\,A\,b-B\,a\right )}{\sqrt {a}\,\left (35\,A\,b-5\,B\,a\right )}\right )\,\left (7\,A\,b-B\,a\right )}{8\,a^{9/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*A)/a + (11*x*(7*A*b - B*a))/(8*a^2) + (5*b^2*x^3*(7*A*b - B*a))/(8*a^4) + (5*b*x^2*(7*A*b - B*a))/(3*a^3
))/(a^3*x^(1/2) + b^3*x^(7/2) + 3*a^2*b*x^(3/2) + 3*a*b^2*x^(5/2)) - (5*atan((5*b^(1/2)*x^(1/2)*(7*A*b - B*a))
/(a^(1/2)*(35*A*b - 5*B*a)))*(7*A*b - B*a))/(8*a^(9/2)*b^(1/2))

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