3.4.69 \(\int \frac {A+B x}{x^{5/2} (a+b x)^3} \, dx\) [369]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {5 \sqrt {b} (7 A b-3 a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {5 (7 A b-3 a B)}{4 a^4 \sqrt {x}}-\frac {5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(5/2)*(a + b*x)^3),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.19, size = 112, normalized size = 0.76 \begin {gather*} \frac {105 A b^3 x^3+a^2 b x (56 A-75 B x)+5 a b^2 x^2 (35 A-9 B x)-8 a^3 (A+3 B x)}{12 a^4 x^{3/2} (a+b x)^2}+\frac {5 \sqrt {b} (7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(5/2)*(a + b*x)^3),x]

[Out]

(105*A*b^3*x^3 + a^2*b*x*(56*A - 75*B*x) + 5*a*b^2*x^2*(35*A - 9*B*x) - 8*a^3*(A + 3*B*x))/(12*a^4*x^(3/2)*(a
+ b*x)^2) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))

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Maple [A]
time = 0.09, size = 101, normalized size = 0.69

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(5/2)/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-2/3*A/a^3/x^(3/2)-2*(-3*A*b+B*a)/a^4/x^(1/2)+2/a^4*b*(((11/8*b^2*A-7/8*a*b*B)*x^(3/2)+1/8*a*(13*A*b-9*B*a)*x^
(1/2))/(b*x+a)^2+5/8*(7*A*b-3*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 = 128, normalized size = 0.87 \begin {gather*} -\frac {8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x}{12 \, {\left (a^{4} b^{2} x^{\frac {7}{2}} + 2 \, a^{5} b x^{\frac {5}{2}} + a^{6} x^{\frac {3}{2}}\right )}} - \frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.19, size = 380, normalized size = 2.59 \begin {gather*} \left [-\frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1703 vs. \(2 (139) = 278\).
time = 89.89, 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}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{a^{3}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b^{3}} & \text {for}\: a = 0 \\- \frac {16 A a^{3} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 A a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 A a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {112 A a^{2} b x \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {210 A a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {210 A a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {350 A a b^{2} x^{2} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 A b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 A b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {210 A b^{3} x^{3} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {45 B a^{3} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {45 B a^{3} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {48 B a^{3} x \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {90 B a^{2} b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {90 B a^{2} b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {150 B a^{2} b x^{2} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {45 B a b^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {45 B a b^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {90 B a b^{2} x^{3} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{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**(5/2)/(b*x+a)**3,x)

[Out]

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

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Giac [A]
time = 1.70, size = 108, normalized size = 0.73 \begin {gather*} -\frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} - \frac {2 \, {\left (3 \, B a x - 9 \, A b x + A a\right )}}{3 \, a^{4} x^{\frac {3}{2}}} - \frac {7 \, B a b^{2} x^{\frac {3}{2}} - 11 \, A b^{3} x^{\frac {3}{2}} + 9 \, B a^{2} b \sqrt {x} - 13 \, A a b^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.45, size = 114, normalized size = 0.78 \begin {gather*} \frac {\frac {2\,x\,\left (7\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {5\,b^2\,x^3\,\left (7\,A\,b-3\,B\,a\right )}{4\,a^4}+\frac {25\,b\,x^2\,\left (7\,A\,b-3\,B\,a\right )}{12\,a^3}}{a^2\,x^{3/2}+b^2\,x^{7/2}+2\,a\,b\,x^{5/2}}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (7\,A\,b-3\,B\,a\right )}{4\,a^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(5/2)*(a + b*x)^3),x)

[Out]

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

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