3.7.95 \(\int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx\) [695]

Optimal. Leaf size=142 \[ \frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}-\frac {5 a^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{7/2}} \]

[Out]

5/3*(-a*d+b*c)*(b*x+a)^(3/2)/c^2/(d*x+c)^(3/2)-(b*x+a)^(5/2)/c/x/(d*x+c)^(3/2)-5*a^(3/2)*(-a*d+b*c)*arctanh(c^
(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(7/2)+5*a*(-a*d+b*c)*(b*x+a)^(1/2)/c^3/(d*x+c)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {96, 95, 214} \begin {gather*} -\frac {5 a^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{7/2}}+\frac {5 a \sqrt {a+b x} (b c-a d)}{c^3 \sqrt {c+d x}}+\frac {5 (a+b x)^{3/2} (b c-a d)}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)/(x^2*(c + d*x)^(5/2)),x]

[Out]

(5*(b*c - a*d)*(a + b*x)^(3/2))/(3*c^2*(c + d*x)^(3/2)) - (a + b*x)^(5/2)/(c*x*(c + d*x)^(3/2)) + (5*a*(b*c -
a*d)*Sqrt[a + b*x])/(c^3*Sqrt[c + d*x]) - (5*a^(3/2)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt
[c + d*x])])/c^(7/2)

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 214

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

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx &=-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {(5 (b c-a d)) \int \frac {(a+b x)^{3/2}}{x (c+d x)^{5/2}} \, dx}{2 c}\\ &=\frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {(5 a (b c-a d)) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{2 c^2}\\ &=\frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}+\frac {\left (5 a^2 (b c-a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 c^3}\\ &=\frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}+\frac {\left (5 a^2 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{c^3}\\ &=\frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}-\frac {5 a^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.50, size = 173, normalized size = 1.22 \begin {gather*} \frac {\sqrt {a+b x} \left (2 b^2 c^2 x^2+2 a b c x (7 c+5 d x)-a^2 \left (3 c^2+20 c d x+15 d^2 x^2\right )\right )}{3 c^3 x (c+d x)^{3/2}}-\frac {5 a^{3/2} \sqrt {\frac {b}{d}} \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \left (-b x+\sqrt {\frac {b}{d}} \sqrt {a+b x} \sqrt {c+d x}\right )}{\sqrt {a} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)/(x^2*(c + d*x)^(5/2)),x]

[Out]

(Sqrt[a + b*x]*(2*b^2*c^2*x^2 + 2*a*b*c*x*(7*c + 5*d*x) - a^2*(3*c^2 + 20*c*d*x + 15*d^2*x^2)))/(3*c^3*x*(c +
d*x)^(3/2)) - (5*a^(3/2)*Sqrt[b/d]*Sqrt[d]*(b*c - a*d)*ArcTanh[(Sqrt[d]*(-(b*x) + Sqrt[b/d]*Sqrt[a + b*x]*Sqrt
[c + d*x]))/(Sqrt[a]*Sqrt[b]*Sqrt[c])])/(Sqrt[b]*c^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(116)=232\).
time = 0.09, size = 502, normalized size = 3.54

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c^{2} d x -15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} x -30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}+20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}+4 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}-40 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x +28 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x -6 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{6 c^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x \sqrt {a c}\, \left (d x +c \right )^{\frac {3}{2}}}\)	\(502\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)/x^2/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(b*x+a)^(1/2)*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*d^3*x^3-15*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b*c*d^2*x^3+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+
c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*c*d^2*x^2-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a
^2*b*c^2*d*x^2+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^3*c^2*d*x-15*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)+2*a*c)/x)*a^2*b*c^3*x-30*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*d^2*
x^2+20*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b*c*d*x^2+4*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*b^2*c^2*x^2-40*(a
*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a^2*c*d*x+28*(a*c)^(1/2)*((d*x+c)*(b*x+a))^(1/2)*a*b*c^2*x-6*((d*x+c)*(b*x+a
))^(1/2)*a^2*c^2*(a*c)^(1/2))/c^3/((d*x+c)*(b*x+a))^(1/2)/x/(a*c)^(1/2)/(d*x+c)^(3/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)/x^2/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (116) = 232\).
time = 1.72, size = 507, normalized size = 3.57 \begin {gather*} \left [-\frac {15 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a^{2} c^{2} - {\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \, {\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}, \frac {15 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (3 \, a^{2} c^{2} - {\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \, {\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)/x^2/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(15*((a*b*c*d^2 - a^2*d^3)*x^3 + 2*(a*b*c^2*d - a^2*c*d^2)*x^2 + (a*b*c^3 - a^2*c^2*d)*x)*sqrt(a/c)*log
((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c
)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(3*a^2*c^2 - (2*b^2*c^2 + 10*a*b*c*d - 15*a^2*d^2)*x^2 - 2*(7*
a*b*c^2 - 10*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^3*d^2*x^3 + 2*c^4*d*x^2 + c^5*x), 1/6*(15*((a*b*c*d^2
- a^2*d^3)*x^3 + 2*(a*b*c^2*d - a^2*c*d^2)*x^2 + (a*b*c^3 - a^2*c^2*d)*x)*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c
+ a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - 2*(3*a^2*c^2 - (2
*b^2*c^2 + 10*a*b*c*d - 15*a^2*d^2)*x^2 - 2*(7*a*b*c^2 - 10*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^3*d^2*
x^3 + 2*c^4*d*x^2 + c^5*x)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)/x**2/(d*x+c)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (116) = 232\).
time = 2.90, size = 633, normalized size = 4.46 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{6} c^{6} d {\left | b \right |} + 4 \, a b^{5} c^{5} d^{2} {\left | b \right |} - 11 \, a^{2} b^{4} c^{4} d^{3} {\left | b \right |} + 6 \, a^{3} b^{3} c^{3} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}} + \frac {6 \, {\left (a b^{6} c^{6} d {\left | b \right |} - 3 \, a^{2} b^{5} c^{5} d^{2} {\left | b \right |} + 3 \, a^{3} b^{4} c^{4} d^{3} {\left | b \right |} - a^{4} b^{3} c^{3} d^{4} {\left | b \right |}\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (\sqrt {b d} a^{2} b^{3} c - \sqrt {b d} a^{3} b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{3} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} a^{2} b^{5} c^{2} - 2 \, \sqrt {b d} a^{3} b^{4} c d + \sqrt {b d} a^{4} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c^{3} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)/x^2/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/3*sqrt(b*x + a)*((b^6*c^6*d*abs(b) + 4*a*b^5*c^5*d^2*abs(b) - 11*a^2*b^4*c^4*d^3*abs(b) + 6*a^3*b^3*c^3*d^4*
abs(b))*(b*x + a)/(b^3*c^7*d - a*b^2*c^6*d^2) + 6*(a*b^6*c^6*d*abs(b) - 3*a^2*b^5*c^5*d^2*abs(b) + 3*a^3*b^4*c
^4*d^3*abs(b) - a^4*b^3*c^3*d^4*abs(b))/(b^3*c^7*d - a*b^2*c^6*d^2))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 5
*(sqrt(b*d)*a^2*b^3*c - sqrt(b*d)*a^3*b^2*d)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b*c^3*abs(b)) - 2*(sqrt(b*d)*a^2*b^5*c^2 -
2*sqrt(b*d)*a^3*b^4*c*d + sqrt(b*d)*a^4*b^3*d^2 - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*
b*d - a*b*d))^2*a^2*b^3*c - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^
2*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)*c^3*abs(b))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^2\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(5/2)/(x^2*(c + d*x)^(5/2)),x)

[Out]

int((a + b*x)^(5/2)/(x^2*(c + d*x)^(5/2)), x)

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