3.4.68 \(\int \frac {A+B x}{x^{3/2} (a+b x)^3} \, dx\) [368]

Optimal. Leaf size=126 \[ -\frac {3 (5 A b-a B)}{4 a^3 b \sqrt {x}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}-\frac {3 (5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 44, 53, 65, 211} \begin {gather*} -\frac {3 (5 A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}-\frac {3 (5 A b-a B)}{4 a^3 b \sqrt {x}}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} (a+b x)}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(5*A*b - a*B))/(4*a^3*b*Sqrt[x]) + (A*b - a*B)/(2*a*b*Sqrt[x]*(a + b*x)^2) + (5*A*b - a*B)/(4*a^2*b*Sqrt[x
]*(a + b*x)) - (3*(5*A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2)*Sqrt[b])

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

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Mathematica [A]
time = 0.17, size = 93, normalized size = 0.74 \begin {gather*} \frac {-15 A b^2 x^2+a b x (-25 A+3 B x)+a^2 (-8 A+5 B x)}{4 a^3 \sqrt {x} (a+b x)^2}+\frac {3 (-5 A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*A*b^2*x^2 + a*b*x*(-25*A + 3*B*x) + a^2*(-8*A + 5*B*x))/(4*a^3*Sqrt[x]*(a + b*x)^2) + (3*(-5*A*b + a*B)*A
rcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2)*Sqrt[b])

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Maple [A]
time = 0.09, size = 84, normalized size = 0.67

method	result	size
derivativedivides	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {7}{8} b^{2} A -\frac {3}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (9 A b -5 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {3 \left (5 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3}}\)	\(84\)
default	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {7}{8} b^{2} A -\frac {3}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (9 A b -5 B a \right ) \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {3 \left (5 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3}}\)	\(84\)
risch	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {7 x^{\frac {3}{2}} b^{2} A}{4 a^{3} \left (b x +a \right )^{2}}+\frac {3 x^{\frac {3}{2}} b B}{4 a^{2} \left (b x +a \right )^{2}}-\frac {9 A \sqrt {x}\, b}{4 a^{2} \left (b x +a \right )^{2}}+\frac {5 B \sqrt {x}}{4 a \left (b x +a \right )^{2}}-\frac {15 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A b}{4 a^{3} \sqrt {a b}}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B}{4 a^{2} \sqrt {a b}}\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 98, normalized size = 0.78 \begin {gather*} -\frac {8 \, A a^{2} - 3 \, {\left (B a b - 5 \, A b^{2}\right )} x^{2} - 5 \, {\left (B a^{2} - 5 \, A a b\right )} x}{4 \, {\left (a^{3} b^{2} x^{\frac {5}{2}} + 2 \, a^{4} b x^{\frac {3}{2}} + a^{5} \sqrt {x}\right )}} + \frac {3 \, {\left (B a - 5 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(8*A*a^2 - 3*(B*a*b - 5*A*b^2)*x^2 - 5*(B*a^2 - 5*A*a*b)*x)/(a^3*b^2*x^(5/2) + 2*a^4*b*x^(3/2) + a^5*sqrt
(x)) + 3/4*(B*a - 5*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3)

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Fricas [A]
time = 1.24, size = 331, normalized size = 2.63 \begin {gather*} \left [\frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} 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 (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}}, -\frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} 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*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1598 vs. \(2 (116) = 232\).
time = 33.42, size = 1598, normalized size = 12.68 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{b^{3}} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\- \frac {15 A a^{2} b \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A a^{2} b \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {16 A a^{2} b \sqrt {- \frac {a}{b}}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {30 A a b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {30 A a b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {50 A a b^{2} x \sqrt {- \frac {a}{b}}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {15 A b^{3} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A b^{3} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {30 A b^{3} x^{2} \sqrt {- \frac {a}{b}}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 B a^{3} \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {3 B a^{3} \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {6 B a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {6 B a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {10 B a^{2} b x \sqrt {- \frac {a}{b}}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 B a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {3 B a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {6 B a b^{2} x^{2} \sqrt {- \frac {a}{b}}}{8 a^{5} b \sqrt {x} \sqrt {- \frac {a}{b}} + 16 a^{4} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 8 a^{3} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2))), Eq(a, 0) & Eq(b, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(5*x**(
5/2)))/b**3, Eq(a, 0)), ((-2*A/sqrt(x) + 2*B*sqrt(x))/a**3, Eq(b, 0)), (-15*A*a**2*b*sqrt(x)*log(sqrt(x) - sqr
t(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) +
15*A*a**2*b*sqrt(x)*log(sqrt(x) + sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b)
+ 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) - 16*A*a**2*b*sqrt(-a/b)/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/
2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) - 30*A*a*b**2*x**(3/2)*log(sqrt(x) - sqrt(-a/b))/(8*a**5*b*sq
rt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) + 30*A*a*b**2*x**(3/2)*
log(sqrt(x) + sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/
2)*sqrt(-a/b)) - 50*A*a*b**2*x*sqrt(-a/b)/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*
a**3*b**3*x**(5/2)*sqrt(-a/b)) - 15*A*b**3*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 1
6*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) + 15*A*b**3*x**(5/2)*log(sqrt(x) + sqrt(-a/
b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) - 30*A*
b**3*x**2*sqrt(-a/b)/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sq
rt(-a/b)) + 3*B*a**3*sqrt(x)*log(sqrt(x) - sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sq
rt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) - 3*B*a**3*sqrt(x)*log(sqrt(x) + sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqr
t(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) + 6*B*a**2*b*x**(3/2)*log(sqrt(x
) - sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a
/b)) - 6*B*a**2*b*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt
(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) + 10*B*a**2*b*x*sqrt(-a/b)/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b*
*2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) + 3*B*a*b**2*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(8*a
**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)) - 3*B*a*b**2*x*
*(5/2)*log(sqrt(x) + sqrt(-a/b))/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-a/b) + 8*a**3*b**3
*x**(5/2)*sqrt(-a/b)) + 6*B*a*b**2*x**2*sqrt(-a/b)/(8*a**5*b*sqrt(x)*sqrt(-a/b) + 16*a**4*b**2*x**(3/2)*sqrt(-
a/b) + 8*a**3*b**3*x**(5/2)*sqrt(-a/b)), True))

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Giac [A]
time = 1.68, size = 86, normalized size = 0.68 \begin {gather*} \frac {3 \, {\left (B a - 5 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} - \frac {2 \, A}{a^{3} \sqrt {x}} + \frac {3 \, B a b x^{\frac {3}{2}} - 7 \, A b^{2} x^{\frac {3}{2}} + 5 \, B a^{2} \sqrt {x} - 9 \, A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.47, size = 116, normalized size = 0.92 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {5\,x\,\left (5\,A\,b-B\,a\right )}{4\,a^2}+\frac {3\,b\,x^2\,\left (5\,A\,b-B\,a\right )}{4\,a^3}}{a^2\,\sqrt {x}+b^2\,x^{5/2}+2\,a\,b\,x^{3/2}}-\frac {3\,\mathrm {atan}\left (\frac {3\,\sqrt {b}\,\sqrt {x}\,\left (5\,A\,b-B\,a\right )}{\sqrt {a}\,\left (15\,A\,b-3\,B\,a\right )}\right )\,\left (5\,A\,b-B\,a\right )}{4\,a^{7/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*A)/a + (5*x*(5*A*b - B*a))/(4*a^2) + (3*b*x^2*(5*A*b - B*a))/(4*a^3))/(a^2*x^(1/2) + b^2*x^(5/2) + 2*a*b
*x^(3/2)) - (3*atan((3*b^(1/2)*x^(1/2)*(5*A*b - B*a))/(a^(1/2)*(15*A*b - 3*B*a)))*(5*A*b - B*a))/(4*a^(7/2)*b^
(1/2))

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