3.2.91 \(\int \frac {\sqrt {x} (A+B x)}{(b x+c x^2)^3} \, dx\) [191]

Optimal. Leaf size=147 \[ \frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {5 (3 b B-7 A c)}{4 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac {5 \sqrt {c} (3 b B-7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{9/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {795, 79, 44, 53, 65, 211} \begin {gather*} -\frac {5 \sqrt {c} (3 b B-7 A c) \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{9/2}}-\frac {5 (3 b B-7 A c)}{4 b^4 \sqrt {x}}+\frac {5 (3 b B-7 A c)}{12 b^3 c x^{3/2}}-\frac {3 b B-7 A c}{4 b^2 c x^{3/2} (b+c x)}-\frac {b B-A c}{2 b c x^{3/2} (b+c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 795

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e^p, Int[(e
*x)^(m + p)*(f + g*x)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m}, x] && IntegerQ[p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.17, size = 117, normalized size = 0.80 \begin {gather*} \frac {-3 b B x \left (8 b^2+25 b c x+15 c^2 x^2\right )+A \left (-8 b^3+56 b^2 c x+175 b c^2 x^2+105 c^3 x^3\right )}{12 b^4 x^{3/2} (b+c x)^2}+\frac {5 \sqrt {c} (-3 b B+7 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*B*x*(8*b^2 + 25*b*c*x + 15*c^2*x^2) + A*(-8*b^3 + 56*b^2*c*x + 175*b*c^2*x^2 + 105*c^3*x^3))/(12*b^4*x^(
3/2)*(b + c*x)^2) + (5*Sqrt[c]*(-3*b*B + 7*A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*b^(9/2))

________________________________________________________________________________________

Maple [A]
time = 0.55, size = 101, normalized size = 0.69

method	result	size
derivativedivides	\(\frac {2 c \left (\frac {\left (\frac {11}{8} A \,c^{2}-\frac {7}{8} b B c \right ) x^{\frac {3}{2}}+\frac {b \left (13 A c -9 B b \right ) \sqrt {x}}{8}}{\left (c x +b \right )^{2}}+\frac {5 \left (7 A c -3 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{b^{4}}-\frac {2 A}{3 b^{3} x^{\frac {3}{2}}}-\frac {2 \left (-3 A c +B b \right )}{b^{4} \sqrt {x}}\)	\(101\)
default	\(\frac {2 c \left (\frac {\left (\frac {11}{8} A \,c^{2}-\frac {7}{8} b B c \right ) x^{\frac {3}{2}}+\frac {b \left (13 A c -9 B b \right ) \sqrt {x}}{8}}{\left (c x +b \right )^{2}}+\frac {5 \left (7 A c -3 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{b^{4}}-\frac {2 A}{3 b^{3} x^{\frac {3}{2}}}-\frac {2 \left (-3 A c +B b \right )}{b^{4} \sqrt {x}}\)	\(101\)
risch	\(-\frac {2 \left (-9 A c x +3 b B x +A b \right )}{3 b^{4} x^{\frac {3}{2}}}+\frac {11 c^{3} x^{\frac {3}{2}} A}{4 b^{4} \left (c x +b \right )^{2}}-\frac {7 c^{2} x^{\frac {3}{2}} B}{4 b^{3} \left (c x +b \right )^{2}}+\frac {13 c^{2} A \sqrt {x}}{4 b^{3} \left (c x +b \right )^{2}}-\frac {9 c B \sqrt {x}}{4 b^{2} \left (c x +b \right )^{2}}+\frac {35 c^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right ) A}{4 b^{4} \sqrt {b c}}-\frac {15 c \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right ) B}{4 b^{3} \sqrt {b c}}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*x^(1/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

2/b^4*c*(((11/8*A*c^2-7/8*b*B*c)*x^(3/2)+1/8*b*(13*A*c-9*B*b)*x^(1/2))/(c*x+b)^2+5/8*(7*A*c-3*B*b)/(b*c)^(1/2)
*arctan(c*x^(1/2)/(b*c)^(1/2)))-2/3*A/b^3/x^(3/2)-2*(-3*A*c+B*b)/b^4/x^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 128, normalized size = 0.87 \begin {gather*} -\frac {8 \, A b^{3} + 15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x}{12 \, {\left (b^{4} c^{2} x^{\frac {7}{2}} + 2 \, b^{5} c x^{\frac {5}{2}} + b^{6} x^{\frac {3}{2}}\right )}} - \frac {5 \, {\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/12*(8*A*b^3 + 15*(3*B*b*c^2 - 7*A*c^3)*x^3 + 25*(3*B*b^2*c - 7*A*b*c^2)*x^2 + 8*(3*B*b^3 - 7*A*b^2*c)*x)/(b
^4*c^2*x^(7/2) + 2*b^5*c*x^(5/2) + b^6*x^(3/2)) - 5/4*(3*B*b*c - 7*A*c^2)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*
c)*b^4)

________________________________________________________________________________________

Fricas [A]
time = 1.43, size = 380, normalized size = 2.59 \begin {gather*} \left [-\frac {15 \, {\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{4} + 2 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{3} + {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x + 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (8 \, A b^{3} + 15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt {x}}{24 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{4} + 2 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{3} + {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x^{2}\right )} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) - {\left (8 \, A b^{3} + 15 \, {\left (3 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 25 \, {\left (3 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 8 \, {\left (3 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt {x}}{12 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[-1/24*(15*((3*B*b*c^2 - 7*A*c^3)*x^4 + 2*(3*B*b^2*c - 7*A*b*c^2)*x^3 + (3*B*b^3 - 7*A*b^2*c)*x^2)*sqrt(-c/b)*
log((c*x + 2*b*sqrt(x)*sqrt(-c/b) - b)/(c*x + b)) + 2*(8*A*b^3 + 15*(3*B*b*c^2 - 7*A*c^3)*x^3 + 25*(3*B*b^2*c
- 7*A*b*c^2)*x^2 + 8*(3*B*b^3 - 7*A*b^2*c)*x)*sqrt(x))/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2), 1/12*(15*((3*B*b
*c^2 - 7*A*c^3)*x^4 + 2*(3*B*b^2*c - 7*A*b*c^2)*x^3 + (3*B*b^3 - 7*A*b^2*c)*x^2)*sqrt(c/b)*arctan(b*sqrt(c/b)/
(c*sqrt(x))) - (8*A*b^3 + 15*(3*B*b*c^2 - 7*A*c^3)*x^3 + 25*(3*B*b^2*c - 7*A*b*c^2)*x^2 + 8*(3*B*b^3 - 7*A*b^2
*c)*x)*sqrt(x))/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1703 vs. \(2 (139) = 278\).
time = 109.95, size = 1703, normalized size = 11.59 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{b^{3}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{c^{3}} & \text {for}\: b = 0 \\- \frac {16 A b^{3} \sqrt {- \frac {b}{c}}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {105 A b^{2} c x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {105 A b^{2} c x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {112 A b^{2} c x \sqrt {- \frac {b}{c}}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {210 A b c^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {210 A b c^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {350 A b c^{2} x^{2} \sqrt {- \frac {b}{c}}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {105 A c^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {105 A c^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {210 A c^{3} x^{3} \sqrt {- \frac {b}{c}}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {45 B b^{3} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {45 B b^{3} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {48 B b^{3} x \sqrt {- \frac {b}{c}}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {90 B b^{2} c x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {90 B b^{2} c x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {150 B b^{2} c x^{2} \sqrt {- \frac {b}{c}}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {45 B b c^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} + \frac {45 B b c^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} - \frac {90 B b c^{2} x^{3} \sqrt {- \frac {b}{c}}}{24 b^{6} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 48 b^{5} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}} + 24 b^{4} c^{2} x^{\frac {7}{2}} \sqrt {- \frac {b}{c}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x**(1/2)/(c*x**2+b*x)**3,x)

[Out]

Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(b, 0) & Eq(c, 0)), ((-2*A/(3*x**(3/2)) - 2*B/sqrt(x)
)/b**3, Eq(c, 0)), ((-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2)))/c**3, Eq(b, 0)), (-16*A*b**3*sqrt(-b/c)/(24*b**6*x*
*(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) + 105*A*b**2*c*x**(3/2)*
log(sqrt(x) - sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)
*sqrt(-b/c)) - 105*A*b**2*c*x**(3/2)*log(sqrt(x) + sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/
2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) + 112*A*b**2*c*x*sqrt(-b/c)/(24*b**6*x**(3/2)*sqrt(-b/c) + 4
8*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) + 210*A*b*c**2*x**(5/2)*log(sqrt(x) - sqrt(-b
/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) - 210*A*
b*c**2*x**(5/2)*log(sqrt(x) + sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b*
*4*c**2*x**(7/2)*sqrt(-b/c)) + 350*A*b*c**2*x**2*sqrt(-b/c)/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*
sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) + 105*A*c**3*x**(7/2)*log(sqrt(x) - sqrt(-b/c))/(24*b**6*x**(3/
2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) - 105*A*c**3*x**(7/2)*log(sq
rt(x) + sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(
-b/c)) + 210*A*c**3*x**3*sqrt(-b/c)/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**
2*x**(7/2)*sqrt(-b/c)) - 45*B*b**3*x**(3/2)*log(sqrt(x) - sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c
*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) + 45*B*b**3*x**(3/2)*log(sqrt(x) + sqrt(-b/c))/(24*b*
*6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) - 48*B*b**3*x*sqrt(
-b/c)/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) - 90*B*
b**2*c*x**(5/2)*log(sqrt(x) - sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b*
*4*c**2*x**(7/2)*sqrt(-b/c)) + 90*B*b**2*c*x**(5/2)*log(sqrt(x) + sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 4
8*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) - 150*B*b**2*c*x**2*sqrt(-b/c)/(24*b**6*x**(3
/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) - 45*B*b*c**2*x**(7/2)*log(
sqrt(x) - sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqr
t(-b/c)) + 45*B*b*c**2*x**(7/2)*log(sqrt(x) + sqrt(-b/c))/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b**5*c*x**(5/2)*sq
rt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)) - 90*B*b*c**2*x**3*sqrt(-b/c)/(24*b**6*x**(3/2)*sqrt(-b/c) + 48*b
**5*c*x**(5/2)*sqrt(-b/c) + 24*b**4*c**2*x**(7/2)*sqrt(-b/c)), True))

________________________________________________________________________________________

Giac [A]
time = 3.19, size = 108, normalized size = 0.73 \begin {gather*} -\frac {5 \, {\left (3 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b^{4}} - \frac {2 \, {\left (3 \, B b x - 9 \, A c x + A b\right )}}{3 \, b^{4} x^{\frac {3}{2}}} - \frac {7 \, B b c^{2} x^{\frac {3}{2}} - 11 \, A c^{3} x^{\frac {3}{2}} + 9 \, B b^{2} c \sqrt {x} - 13 \, A b c^{2} \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-5/4*(3*B*b*c - 7*A*c^2)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^4) - 2/3*(3*B*b*x - 9*A*c*x + A*b)/(b^4*x^(3
/2)) - 1/4*(7*B*b*c^2*x^(3/2) - 11*A*c^3*x^(3/2) + 9*B*b^2*c*sqrt(x) - 13*A*b*c^2*sqrt(x))/((c*x + b)^2*b^4)

________________________________________________________________________________________

Mupad [B]
time = 1.11, size = 114, normalized size = 0.78 \begin {gather*} \frac {\frac {2\,x\,\left (7\,A\,c-3\,B\,b\right )}{3\,b^2}-\frac {2\,A}{3\,b}+\frac {5\,c^2\,x^3\,\left (7\,A\,c-3\,B\,b\right )}{4\,b^4}+\frac {25\,c\,x^2\,\left (7\,A\,c-3\,B\,b\right )}{12\,b^3}}{b^2\,x^{3/2}+c^2\,x^{7/2}+2\,b\,c\,x^{5/2}}+\frac {5\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (7\,A\,c-3\,B\,b\right )}{4\,b^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2)*(A + B*x))/(b*x + c*x^2)^3,x)

[Out]

((2*x*(7*A*c - 3*B*b))/(3*b^2) - (2*A)/(3*b) + (5*c^2*x^3*(7*A*c - 3*B*b))/(4*b^4) + (25*c*x^2*(7*A*c - 3*B*b)
)/(12*b^3))/(b^2*x^(3/2) + c^2*x^(7/2) + 2*b*c*x^(5/2)) + (5*c^(1/2)*atan((c^(1/2)*x^(1/2))/b^(1/2))*(7*A*c -
3*B*b))/(4*b^(9/2))

________________________________________________________________________________________