\(\int \frac {1}{(\frac {c}{(a+b x)^3})^{5/2}} \, dx\) [2836]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\frac {2 (a+b x)^7}{17 b c^2 \sqrt {\frac {c}{(a+b x)^3}}} \]

[Out]

2/17*(b*x+a)^7/b/c^2/(c/(b*x+a)^3)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 30} \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\frac {2 (a+b x)^7}{17 b c^2 \sqrt {\frac {c}{(a+b x)^3}}} \]

[In]

Int[(c/(a + b*x)^3)^(-5/2),x]

[Out]

(2*(a + b*x)^7)/(17*b*c^2*Sqrt[c/(a + b*x)^3])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 253

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (\frac {c}{x^3}\right )^{5/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int x^{15/2} \, dx,x,a+b x\right )}{b c^2 \sqrt {\frac {c}{(a+b x)^3}} (a+b x)^{3/2}} \\ & = \frac {2 (a+b x)^7}{17 b c^2 \sqrt {\frac {c}{(a+b x)^3}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\frac {2 (a+b x)}{17 b \left (\frac {c}{(a+b x)^3}\right )^{5/2}} \]

[In]

Integrate[(c/(a + b*x)^3)^(-5/2),x]

[Out]

(2*(a + b*x))/(17*b*(c/(a + b*x)^3)^(5/2))

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73

method	result	size
gosper	\(\frac {\frac {2 b x}{17}+\frac {2 a}{17}}{b \left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {5}{2}}}\)	\(22\)
default	\(\frac {\frac {2 b x}{17}+\frac {2 a}{17}}{b \left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {5}{2}}}\)	\(22\)
risch	\(\frac {\frac {2}{17} a^{8}+\frac {16}{17} a^{7} x b +\frac {56}{17} a^{6} x^{2} b^{2}+\frac {112}{17} a^{5} b^{3} x^{3}+\frac {140}{17} a^{4} x^{4} b^{4}+\frac {112}{17} a^{3} x^{5} b^{5}+\frac {56}{17} a^{2} x^{6} b^{6}+\frac {16}{17} a \,x^{7} b^{7}+\frac {2}{17} b^{8} x^{8}}{c^{2} \left (b x +a \right ) \sqrt {\frac {c}{\left (b x +a \right )^{3}}}\, b}\)	\(109\)
trager	\(\frac {2 \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b^{8} x^{8}+8 a \,x^{7} b^{7}+28 a^{2} x^{6} b^{6}+56 a^{3} x^{5} b^{5}+70 a^{4} x^{4} b^{4}+56 a^{5} b^{3} x^{3}+28 a^{6} x^{2} b^{2}+8 a^{7} x b +a^{8}\right ) \sqrt {\frac {c}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}}}{17 c^{3} b}\)	\(140\)

[In]

int(1/(c/(b*x+a)^3)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/17*(b*x+a)/b/(c/(b*x+a)^3)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.83 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\frac {2 \, {\left (b^{10} x^{10} + 10 \, a b^{9} x^{9} + 45 \, a^{2} b^{8} x^{8} + 120 \, a^{3} b^{7} x^{7} + 210 \, a^{4} b^{6} x^{6} + 252 \, a^{5} b^{5} x^{5} + 210 \, a^{6} b^{4} x^{4} + 120 \, a^{7} b^{3} x^{3} + 45 \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10}\right )} \sqrt {\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{17 \, b c^{3}} \]

[In]

integrate(1/(c/(b*x+a)^3)^(5/2),x, algorithm="fricas")

[Out]

2/17*(b^10*x^10 + 10*a*b^9*x^9 + 45*a^2*b^8*x^8 + 120*a^3*b^7*x^7 + 210*a^4*b^6*x^6 + 252*a^5*b^5*x^5 + 210*a^
6*b^4*x^4 + 120*a^7*b^3*x^3 + 45*a^8*b^2*x^2 + 10*a^9*b*x + a^10)*sqrt(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x +
a^3))/(b*c^3)

Sympy [A] (verification not implemented)

Time = 1.95 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {a}{b} + x\right )}{17 \left (\frac {c}{\left (a + b x\right )^{3}}\right )^{\frac {5}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\frac {c}{a^{3}}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(c/(b*x+a)**3)**(5/2),x)

[Out]

Piecewise((2*(a/b + x)/(17*(c/(a + b*x)**3)**(5/2)), Ne(b, 0)), (x/(c/a**3)**(5/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\frac {2 \, {\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{\frac {15}{2}}}{17 \, b c^{3}} \]

[In]

integrate(1/(c/(b*x+a)^3)^(5/2),x, algorithm="maxima")

[Out]

2/17*(b*sqrt(c)*x + a*sqrt(c))*(b*x + a)^(15/2)/(b*c^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (26) = 52\).

Time = 0.30 (sec) , antiderivative size = 840, normalized size of antiderivative = 28.00 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\frac {2 \, {\left (109395 \, \sqrt {b c x + a c} a^{8} - \frac {291720 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{7}}{c} + \frac {204204 \, {\left (15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}\right )} a^{6}}{c^{2}} - \frac {175032 \, {\left (35 \, \sqrt {b c x + a c} a^{3} c^{3} - 35 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{2} c^{2} + 21 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a c - 5 \, {\left (b c x + a c\right )}^{\frac {7}{2}}\right )} a^{5}}{c^{3}} + \frac {24310 \, {\left (315 \, \sqrt {b c x + a c} a^{4} c^{4} - 420 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{3} c^{3} + 378 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{2} c^{2} - 180 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a c + 35 \, {\left (b c x + a c\right )}^{\frac {9}{2}}\right )} a^{4}}{c^{4}} - \frac {8840 \, {\left (693 \, \sqrt {b c x + a c} a^{5} c^{5} - 1155 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{4} c^{4} + 1386 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{3} c^{3} - 990 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{2} c^{2} + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a c - 63 \, {\left (b c x + a c\right )}^{\frac {11}{2}}\right )} a^{3}}{c^{5}} + \frac {1020 \, {\left (3003 \, \sqrt {b c x + a c} a^{6} c^{6} - 6006 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{5} c^{5} + 9009 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{4} c^{4} - 8580 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{3} c^{3} + 5005 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{2} c^{2} - 1638 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a c + 231 \, {\left (b c x + a c\right )}^{\frac {13}{2}}\right )} a^{2}}{c^{6}} - \frac {136 \, {\left (6435 \, \sqrt {b c x + a c} a^{7} c^{7} - 15015 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{6} c^{6} + 27027 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{5} c^{5} - 32175 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{4} c^{4} + 25025 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{3} c^{3} - 12285 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a^{2} c^{2} + 3465 \, {\left (b c x + a c\right )}^{\frac {13}{2}} a c - 429 \, {\left (b c x + a c\right )}^{\frac {15}{2}}\right )} a}{c^{7}} + \frac {109395 \, \sqrt {b c x + a c} a^{8} c^{8} - 291720 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{7} c^{7} + 612612 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{6} c^{6} - 875160 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{5} c^{5} + 850850 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{4} c^{4} - 556920 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a^{3} c^{3} + 235620 \, {\left (b c x + a c\right )}^{\frac {13}{2}} a^{2} c^{2} - 58344 \, {\left (b c x + a c\right )}^{\frac {15}{2}} a c + 6435 \, {\left (b c x + a c\right )}^{\frac {17}{2}}}{c^{8}}\right )}}{109395 \, b c^{3} \mathrm {sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm {sgn}\left (b x + a\right )} \]

[In]

integrate(1/(c/(b*x+a)^3)^(5/2),x, algorithm="giac")

[Out]

2/109395*(109395*sqrt(b*c*x + a*c)*a^8 - 291720*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^7/c + 204204
*(15*sqrt(b*c*x + a*c)*a^2*c^2 - 10*(b*c*x + a*c)^(3/2)*a*c + 3*(b*c*x + a*c)^(5/2))*a^6/c^2 - 175032*(35*sqrt
(b*c*x + a*c)*a^3*c^3 - 35*(b*c*x + a*c)^(3/2)*a^2*c^2 + 21*(b*c*x + a*c)^(5/2)*a*c - 5*(b*c*x + a*c)^(7/2))*a
^5/c^3 + 24310*(315*sqrt(b*c*x + a*c)*a^4*c^4 - 420*(b*c*x + a*c)^(3/2)*a^3*c^3 + 378*(b*c*x + a*c)^(5/2)*a^2*
c^2 - 180*(b*c*x + a*c)^(7/2)*a*c + 35*(b*c*x + a*c)^(9/2))*a^4/c^4 - 8840*(693*sqrt(b*c*x + a*c)*a^5*c^5 - 11
55*(b*c*x + a*c)^(3/2)*a^4*c^4 + 1386*(b*c*x + a*c)^(5/2)*a^3*c^3 - 990*(b*c*x + a*c)^(7/2)*a^2*c^2 + 385*(b*c
*x + a*c)^(9/2)*a*c - 63*(b*c*x + a*c)^(11/2))*a^3/c^5 + 1020*(3003*sqrt(b*c*x + a*c)*a^6*c^6 - 6006*(b*c*x +
a*c)^(3/2)*a^5*c^5 + 9009*(b*c*x + a*c)^(5/2)*a^4*c^4 - 8580*(b*c*x + a*c)^(7/2)*a^3*c^3 + 5005*(b*c*x + a*c)^
(9/2)*a^2*c^2 - 1638*(b*c*x + a*c)^(11/2)*a*c + 231*(b*c*x + a*c)^(13/2))*a^2/c^6 - 136*(6435*sqrt(b*c*x + a*c
)*a^7*c^7 - 15015*(b*c*x + a*c)^(3/2)*a^6*c^6 + 27027*(b*c*x + a*c)^(5/2)*a^5*c^5 - 32175*(b*c*x + a*c)^(7/2)*
a^4*c^4 + 25025*(b*c*x + a*c)^(9/2)*a^3*c^3 - 12285*(b*c*x + a*c)^(11/2)*a^2*c^2 + 3465*(b*c*x + a*c)^(13/2)*a
*c - 429*(b*c*x + a*c)^(15/2))*a/c^7 + (109395*sqrt(b*c*x + a*c)*a^8*c^8 - 291720*(b*c*x + a*c)^(3/2)*a^7*c^7
+ 612612*(b*c*x + a*c)^(5/2)*a^6*c^6 - 875160*(b*c*x + a*c)^(7/2)*a^5*c^5 + 850850*(b*c*x + a*c)^(9/2)*a^4*c^4
- 556920*(b*c*x + a*c)^(11/2)*a^3*c^3 + 235620*(b*c*x + a*c)^(13/2)*a^2*c^2 - 58344*(b*c*x + a*c)^(15/2)*a*c
+ 6435*(b*c*x + a*c)^(17/2))/c^8)/(b*c^3*sgn(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sgn(b*x + a))

Mupad [B] (verification not implemented)

Time = 6.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 5.07 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^3}\right )^{5/2}} \, dx=\sqrt {\frac {c}{{\left (a+b\,x\right )}^3}}\,\left (\frac {20\,a^9\,x}{17\,c^3}+\frac {2\,a^{10}}{17\,b\,c^3}+\frac {2\,b^9\,x^{10}}{17\,c^3}+\frac {90\,a^8\,b\,x^2}{17\,c^3}+\frac {20\,a\,b^8\,x^9}{17\,c^3}+\frac {240\,a^7\,b^2\,x^3}{17\,c^3}+\frac {420\,a^6\,b^3\,x^4}{17\,c^3}+\frac {504\,a^5\,b^4\,x^5}{17\,c^3}+\frac {420\,a^4\,b^5\,x^6}{17\,c^3}+\frac {240\,a^3\,b^6\,x^7}{17\,c^3}+\frac {90\,a^2\,b^7\,x^8}{17\,c^3}\right ) \]

[In]

int(1/(c/(a + b*x)^3)^(5/2),x)

[Out]

(c/(a + b*x)^3)^(1/2)*((20*a^9*x)/(17*c^3) + (2*a^10)/(17*b*c^3) + (2*b^9*x^10)/(17*c^3) + (90*a^8*b*x^2)/(17*
c^3) + (20*a*b^8*x^9)/(17*c^3) + (240*a^7*b^2*x^3)/(17*c^3) + (420*a^6*b^3*x^4)/(17*c^3) + (504*a^5*b^4*x^5)/(
17*c^3) + (420*a^4*b^5*x^6)/(17*c^3) + (240*a^3*b^6*x^7)/(17*c^3) + (90*a^2*b^7*x^8)/(17*c^3))