\(\int \frac {(c+d x)^2}{x^2 (a x+b x^2)^{5/2}} \, dx\) [183]

Optimal result

Integrand size = 24, antiderivative size = 238 \[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=\frac {2 \left (35 d^2+\frac {16 b c (5 b c-7 a d)}{a^2}\right )}{105 a \left (a x+b x^2\right )^{3/2}}-\frac {2 c^2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}+\frac {4 c (5 b c-7 a d)}{35 a^2 x \left (a x+b x^2\right )^{3/2}}+\frac {4 \left (35 a^2 d^2+16 b c (5 b c-7 a d)\right )}{35 a^4 x \sqrt {a x+b x^2}}-\frac {16 \left (35 a^2 d^2+16 b c (5 b c-7 a d)\right ) \sqrt {a x+b x^2}}{105 a^5 x^2}+\frac {32 b \left (35 a^2 d^2+16 b c (5 b c-7 a d)\right ) \sqrt {a x+b x^2}}{105 a^6 x} \] Output:

2/105*(35*d^2+16*b*c*(-7*a*d+5*b*c)/a^2)/a/(b*x^2+a*x)^(3/2)-2/7*c^2/a/x^2 
/(b*x^2+a*x)^(3/2)+4/35*c*(-7*a*d+5*b*c)/a^2/x/(b*x^2+a*x)^(3/2)+4/35*(35* 
a^2*d^2+16*b*c*(-7*a*d+5*b*c))/a^4/x/(b*x^2+a*x)^(1/2)-16/105*(35*a^2*d^2+ 
16*b*c*(-7*a*d+5*b*c))*(b*x^2+a*x)^(1/2)/a^5/x^2+32/105*b*(35*a^2*d^2+16*b 
*c*(-7*a*d+5*b*c))*(b*x^2+a*x)^(1/2)/a^6/x
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=\frac {2560 b^5 c^2 x^5+256 a b^4 c x^4 (15 c-14 d x)+32 a^2 b^3 x^3 \left (30 c^2-168 c d x+35 d^2 x^2\right )-2 a^5 \left (15 c^2+42 c d x+35 d^2 x^2\right )+16 a^3 b^2 x^2 \left (-10 c^2-84 c d x+105 d^2 x^2\right )+4 a^4 b x \left (15 c^2+56 c d x+105 d^2 x^2\right )}{105 a^6 x^2 (x (a+b x))^{3/2}} \] Input:

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

Output:

(2560*b^5*c^2*x^5 + 256*a*b^4*c*x^4*(15*c - 14*d*x) + 32*a^2*b^3*x^3*(30*c 
^2 - 168*c*d*x + 35*d^2*x^2) - 2*a^5*(15*c^2 + 42*c*d*x + 35*d^2*x^2) + 16 
*a^3*b^2*x^2*(-10*c^2 - 84*c*d*x + 105*d^2*x^2) + 4*a^4*b*x*(15*c^2 + 56*c 
*d*x + 105*d^2*x^2))/(105*a^6*x^2*(x*(a + b*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.75, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1262, 27, 1220, 1129, 1089, 1088}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1262

\(\displaystyle -\frac {\int -\frac {8 b c^2+d (16 b c-5 a d) x}{2 x^2 \left (b x^2+a x\right )^{5/2}}dx}{4 b}-\frac {d^2}{4 b x \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {8 b c^2+d (16 b c-5 a d) x}{x^2 \left (b x^2+a x\right )^{5/2}}dx}{8 b}-\frac {d^2}{4 b x \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {-\frac {\left (35 a^2 d^2-112 a b c d+80 b^2 c^2\right ) \int \frac {1}{x \left (b x^2+a x\right )^{5/2}}dx}{7 a}-\frac {16 b c^2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}}{8 b}-\frac {d^2}{4 b x \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {-\frac {\left (35 a^2 d^2-112 a b c d+80 b^2 c^2\right ) \left (-\frac {8 b \int \frac {1}{\left (b x^2+a x\right )^{5/2}}dx}{5 a}-\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}}\right )}{7 a}-\frac {16 b c^2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}}{8 b}-\frac {d^2}{4 b x \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1089

\(\displaystyle \frac {-\frac {\left (35 a^2 d^2-112 a b c d+80 b^2 c^2\right ) \left (-\frac {8 b \left (-\frac {8 b \int \frac {1}{\left (b x^2+a x\right )^{3/2}}dx}{3 a^2}-\frac {2 (a+2 b x)}{3 a^2 \left (a x+b x^2\right )^{3/2}}\right )}{5 a}-\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}}\right )}{7 a}-\frac {16 b c^2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}}{8 b}-\frac {d^2}{4 b x \left (a x+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1088

\(\displaystyle \frac {-\frac {\left (-\frac {8 b \left (\frac {16 b (a+2 b x)}{3 a^4 \sqrt {a x+b x^2}}-\frac {2 (a+2 b x)}{3 a^2 \left (a x+b x^2\right )^{3/2}}\right )}{5 a}-\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}}\right ) \left (35 a^2 d^2-112 a b c d+80 b^2 c^2\right )}{7 a}-\frac {16 b c^2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}}{8 b}-\frac {d^2}{4 b x \left (a x+b x^2\right )^{3/2}}\)

Input:

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

Output:

-1/4*d^2/(b*x*(a*x + b*x^2)^(3/2)) + ((-16*b*c^2)/(7*a*x^2*(a*x + b*x^2)^( 
3/2)) - ((80*b^2*c^2 - 112*a*b*c*d + 35*a^2*d^2)*(-2/(5*a*x*(a*x + b*x^2)^ 
(3/2)) - (8*b*((-2*(a + 2*b*x))/(3*a^2*(a*x + b*x^2)^(3/2)) + (16*b*(a + 2 
*b*x))/(3*a^4*Sqrt[a*x + b*x^2])))/(5*a)))/(7*a))/(8*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1088

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 
2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0]
 

rule 1089

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1129

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) 
))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d 
, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 
2], 0]
 

rule 1220

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

rule 1262

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {\left (-70 d^{2} x^{2}-84 c d x -30 c^{2}\right ) a^{5}+60 \left (7 d^{2} x^{2}+\frac {56}{15} c d x +c^{2}\right ) x b \,a^{4}-160 x^{2} \left (-\frac {21}{2} d^{2} x^{2}+\frac {42}{5} c d x +c^{2}\right ) b^{2} a^{3}+960 x^{3} \left (\frac {7}{6} d^{2} x^{2}-\frac {28}{5} c d x +c^{2}\right ) b^{3} a^{2}+3840 \left (-\frac {14 d x}{15}+c \right ) x^{4} b^{4} c a +2560 b^{5} c^{2} x^{5}}{105 \sqrt {x \left (b x +a \right )}\, x^{3} \left (b x +a \right ) a^{6}}\)	\(156\)
risch	\(-\frac {2 \left (b x +a \right ) \left (-280 d^{2} x^{3} a^{2} b +1022 a \,b^{2} c d \,x^{3}-790 b^{3} c^{2} x^{3}+35 a^{3} d^{2} x^{2}-196 x^{2} a^{2} b c d +185 a \,b^{2} c^{2} x^{2}+42 a^{3} c d x -60 a^{2} b \,c^{2} x +15 c^{2} a^{3}\right )}{105 a^{6} x^{3} \sqrt {x \left (b x +a \right )}}+\frac {2 x \left (8 a b d x -14 b^{2} c x +9 a^{2} d -15 a b c \right ) \left (a d -b c \right ) b^{2}}{3 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right ) a^{6}}\)	\(177\)
gosper	\(-\frac {2 \left (b x +a \right ) \left (-560 a^{2} b^{3} d^{2} x^{5}+1792 a \,b^{4} c d \,x^{5}-1280 b^{5} c^{2} x^{5}-840 a^{3} b^{2} d^{2} x^{4}+2688 a^{2} b^{3} c d \,x^{4}-1920 a \,b^{4} c^{2} x^{4}-210 a^{4} b \,d^{2} x^{3}+672 a^{3} b^{2} c d \,x^{3}-480 a^{2} b^{3} c^{2} x^{3}+35 a^{5} d^{2} x^{2}-112 a^{4} b c d \,x^{2}+80 a^{3} b^{2} c^{2} x^{2}+42 a^{5} c d x -30 a^{4} b \,c^{2} x +15 c^{2} a^{5}\right )}{105 x \,a^{6} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}\)	\(202\)
orering	\(-\frac {2 \left (b x +a \right ) \left (-560 a^{2} b^{3} d^{2} x^{5}+1792 a \,b^{4} c d \,x^{5}-1280 b^{5} c^{2} x^{5}-840 a^{3} b^{2} d^{2} x^{4}+2688 a^{2} b^{3} c d \,x^{4}-1920 a \,b^{4} c^{2} x^{4}-210 a^{4} b \,d^{2} x^{3}+672 a^{3} b^{2} c d \,x^{3}-480 a^{2} b^{3} c^{2} x^{3}+35 a^{5} d^{2} x^{2}-112 a^{4} b c d \,x^{2}+80 a^{3} b^{2} c^{2} x^{2}+42 a^{5} c d x -30 a^{4} b \,c^{2} x +15 c^{2} a^{5}\right )}{105 x \,a^{6} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}\)	\(202\)
trager	\(-\frac {2 \left (-560 a^{2} b^{3} d^{2} x^{5}+1792 a \,b^{4} c d \,x^{5}-1280 b^{5} c^{2} x^{5}-840 a^{3} b^{2} d^{2} x^{4}+2688 a^{2} b^{3} c d \,x^{4}-1920 a \,b^{4} c^{2} x^{4}-210 a^{4} b \,d^{2} x^{3}+672 a^{3} b^{2} c d \,x^{3}-480 a^{2} b^{3} c^{2} x^{3}+35 a^{5} d^{2} x^{2}-112 a^{4} b c d \,x^{2}+80 a^{3} b^{2} c^{2} x^{2}+42 a^{5} c d x -30 a^{4} b \,c^{2} x +15 c^{2} a^{5}\right ) \sqrt {b \,x^{2}+a x}}{105 a^{6} x^{4} \left (b x +a \right )^{2}}\)	\(204\)
default	\(d^{2} \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )+c^{2} \left (-\frac {2}{7 a \,x^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {10 b \left (-\frac {2}{5 a x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {8 b \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{5 a}\right )}{7 a}\right )+2 c d \left (-\frac {2}{5 a x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {8 b \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{5 a}\right )\)	\(230\)

Input:

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

Output:

1/105*((-70*d^2*x^2-84*c*d*x-30*c^2)*a^5+60*(7*d^2*x^2+56/15*c*d*x+c^2)*x* 
b*a^4-160*x^2*(-21/2*d^2*x^2+42/5*c*d*x+c^2)*b^2*a^3+960*x^3*(7/6*d^2*x^2- 
28/5*c*d*x+c^2)*b^3*a^2+3840*(-14/15*d*x+c)*x^4*b^4*c*a+2560*b^5*c^2*x^5)/ 
(x*(b*x+a))^(1/2)/x^3/(b*x+a)/a^6
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (15 \, a^{5} c^{2} - 16 \, {\left (80 \, b^{5} c^{2} - 112 \, a b^{4} c d + 35 \, a^{2} b^{3} d^{2}\right )} x^{5} - 24 \, {\left (80 \, a b^{4} c^{2} - 112 \, a^{2} b^{3} c d + 35 \, a^{3} b^{2} d^{2}\right )} x^{4} - 6 \, {\left (80 \, a^{2} b^{3} c^{2} - 112 \, a^{3} b^{2} c d + 35 \, a^{4} b d^{2}\right )} x^{3} + {\left (80 \, a^{3} b^{2} c^{2} - 112 \, a^{4} b c d + 35 \, a^{5} d^{2}\right )} x^{2} - 6 \, {\left (5 \, a^{4} b c^{2} - 7 \, a^{5} c d\right )} x\right )} \sqrt {b x^{2} + a x}}{105 \, {\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} \] Input:

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

Output:

-2/105*(15*a^5*c^2 - 16*(80*b^5*c^2 - 112*a*b^4*c*d + 35*a^2*b^3*d^2)*x^5 
- 24*(80*a*b^4*c^2 - 112*a^2*b^3*c*d + 35*a^3*b^2*d^2)*x^4 - 6*(80*a^2*b^3 
*c^2 - 112*a^3*b^2*c*d + 35*a^4*b*d^2)*x^3 + (80*a^3*b^2*c^2 - 112*a^4*b*c 
*d + 35*a^5*d^2)*x^2 - 6*(5*a^4*b*c^2 - 7*a^5*c*d)*x)*sqrt(b*x^2 + a*x)/(a 
^6*b^2*x^6 + 2*a^7*b*x^5 + a^8*x^4)
 

Sympy [F]

\[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x\right )^{2}}{x^{2} \left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.36 \[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=-\frac {64 \, b^{3} c^{2} x}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} + \frac {512 \, b^{4} c^{2} x}{21 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {64 \, b^{2} c d x}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} - \frac {512 \, b^{3} c d x}{15 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {4 \, b d^{2} x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} + \frac {32 \, b^{2} d^{2} x}{3 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {32 \, b^{2} c^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} + \frac {256 \, b^{3} c^{2}}{21 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {32 \, b c d}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} - \frac {256 \, b^{2} c d}{15 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {2 \, d^{2}}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a} + \frac {16 \, b d^{2}}{3 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {4 \, b c^{2}}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x} - \frac {4 \, c d}{5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x} - \frac {2 \, c^{2}}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{2}} \] Input:

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

Output:

-64/21*b^3*c^2*x/((b*x^2 + a*x)^(3/2)*a^4) + 512/21*b^4*c^2*x/(sqrt(b*x^2 
+ a*x)*a^6) + 64/15*b^2*c*d*x/((b*x^2 + a*x)^(3/2)*a^3) - 512/15*b^3*c*d*x 
/(sqrt(b*x^2 + a*x)*a^5) - 4/3*b*d^2*x/((b*x^2 + a*x)^(3/2)*a^2) + 32/3*b^ 
2*d^2*x/(sqrt(b*x^2 + a*x)*a^4) - 32/21*b^2*c^2/((b*x^2 + a*x)^(3/2)*a^3) 
+ 256/21*b^3*c^2/(sqrt(b*x^2 + a*x)*a^5) + 32/15*b*c*d/((b*x^2 + a*x)^(3/2 
)*a^2) - 256/15*b^2*c*d/(sqrt(b*x^2 + a*x)*a^4) - 2/3*d^2/((b*x^2 + a*x)^( 
3/2)*a) + 16/3*b*d^2/(sqrt(b*x^2 + a*x)*a^3) + 4/7*b*c^2/((b*x^2 + a*x)^(3 
/2)*a^2*x) - 4/5*c*d/((b*x^2 + a*x)^(3/2)*a*x) - 2/7*c^2/((b*x^2 + a*x)^(3 
/2)*a*x^2)
 

Giac [F]

\[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b x^{2} + a x\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.98 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.24 \[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=\frac {\left (\frac {560\,a^2\,b\,d^2-1792\,a\,b^2\,c\,d+1280\,b^3\,c^2}{105\,a^5}+\frac {2\,b\,x\,\left (560\,a^2\,b\,d^2-1792\,a\,b^2\,c\,d+1280\,b^3\,c^2\right )}{105\,a^6}\right )\,\sqrt {b\,x^2+a\,x}}{x\,\left (a+b\,x\right )}-\frac {2\,c^2\,\sqrt {b\,x^2+a\,x}}{7\,a^3\,x^4}-\frac {\sqrt {b\,x^2+a\,x}\,\left (\frac {70\,a^4\,d^2-392\,a^3\,b\,c\,d+370\,a^2\,b^2\,c^2}{105\,a^5}-x\,\left (\frac {a\,\left (\frac {16\,b^3\,c\,\left (7\,a\,d-10\,b\,c\right )}{105\,a^5}-\frac {8\,b^3\,c\,\left (91\,a\,d-115\,b\,c\right )}{105\,a^5}\right )}{b}-\frac {2\,b\,\left (70\,a^4\,d^2-392\,a^3\,b\,c\,d+370\,a^2\,b^2\,c^2\right )}{105\,a^6}+\frac {4\,b^2\,c\,\left (91\,a\,d-115\,b\,c\right )}{105\,a^4}\right )\right )}{x^2\,{\left (a+b\,x\right )}^2}-\frac {4\,c\,\sqrt {b\,x^2+a\,x}\,\left (7\,a\,d-10\,b\,c\right )}{35\,a^4\,x^3} \] Input:

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

Output:

(((1280*b^3*c^2 + 560*a^2*b*d^2 - 1792*a*b^2*c*d)/(105*a^5) + (2*b*x*(1280 
*b^3*c^2 + 560*a^2*b*d^2 - 1792*a*b^2*c*d))/(105*a^6))*(a*x + b*x^2)^(1/2) 
)/(x*(a + b*x)) - (2*c^2*(a*x + b*x^2)^(1/2))/(7*a^3*x^4) - ((a*x + b*x^2) 
^(1/2)*((70*a^4*d^2 + 370*a^2*b^2*c^2 - 392*a^3*b*c*d)/(105*a^5) - x*((a*( 
(16*b^3*c*(7*a*d - 10*b*c))/(105*a^5) - (8*b^3*c*(91*a*d - 115*b*c))/(105* 
a^5)))/b - (2*b*(70*a^4*d^2 + 370*a^2*b^2*c^2 - 392*a^3*b*c*d))/(105*a^6) 
+ (4*b^2*c*(91*a*d - 115*b*c))/(105*a^4))))/(x^2*(a + b*x)^2) - (4*c*(a*x 
+ b*x^2)^(1/2)*(7*a*d - 10*b*c))/(35*a^4*x^3)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.47 \[ \int \frac {(c+d x)^2}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx=\frac {-\frac {32 \sqrt {b}\, \sqrt {b x +a}\, a^{3} b \,d^{2} x^{4}}{3}+\frac {512 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{2} c d \,x^{4}}{15}-\frac {32 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{2} d^{2} x^{5}}{3}-\frac {512 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{3} c^{2} x^{4}}{21}+\frac {512 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{3} c d \,x^{5}}{15}-\frac {512 \sqrt {b}\, \sqrt {b x +a}\, b^{4} c^{2} x^{5}}{21}-\frac {2 \sqrt {x}\, a^{5} c^{2}}{7}-\frac {4 \sqrt {x}\, a^{5} c d x}{5}-\frac {2 \sqrt {x}\, a^{5} d^{2} x^{2}}{3}+\frac {4 \sqrt {x}\, a^{4} b \,c^{2} x}{7}+\frac {32 \sqrt {x}\, a^{4} b c d \,x^{2}}{15}+4 \sqrt {x}\, a^{4} b \,d^{2} x^{3}-\frac {32 \sqrt {x}\, a^{3} b^{2} c^{2} x^{2}}{21}-\frac {64 \sqrt {x}\, a^{3} b^{2} c d \,x^{3}}{5}+16 \sqrt {x}\, a^{3} b^{2} d^{2} x^{4}+\frac {64 \sqrt {x}\, a^{2} b^{3} c^{2} x^{3}}{7}-\frac {256 \sqrt {x}\, a^{2} b^{3} c d \,x^{4}}{5}+\frac {32 \sqrt {x}\, a^{2} b^{3} d^{2} x^{5}}{3}+\frac {256 \sqrt {x}\, a \,b^{4} c^{2} x^{4}}{7}-\frac {512 \sqrt {x}\, a \,b^{4} c d \,x^{5}}{15}+\frac {512 \sqrt {x}\, b^{5} c^{2} x^{5}}{21}}{\sqrt {b x +a}\, a^{6} x^{4} \left (b x +a \right )} \] Input:

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

Output:

(2*( - 560*sqrt(b)*sqrt(a + b*x)*a**3*b*d**2*x**4 + 1792*sqrt(b)*sqrt(a + 
b*x)*a**2*b**2*c*d*x**4 - 560*sqrt(b)*sqrt(a + b*x)*a**2*b**2*d**2*x**5 - 
1280*sqrt(b)*sqrt(a + b*x)*a*b**3*c**2*x**4 + 1792*sqrt(b)*sqrt(a + b*x)*a 
*b**3*c*d*x**5 - 1280*sqrt(b)*sqrt(a + b*x)*b**4*c**2*x**5 - 15*sqrt(x)*a* 
*5*c**2 - 42*sqrt(x)*a**5*c*d*x - 35*sqrt(x)*a**5*d**2*x**2 + 30*sqrt(x)*a 
**4*b*c**2*x + 112*sqrt(x)*a**4*b*c*d*x**2 + 210*sqrt(x)*a**4*b*d**2*x**3 
- 80*sqrt(x)*a**3*b**2*c**2*x**2 - 672*sqrt(x)*a**3*b**2*c*d*x**3 + 840*sq 
rt(x)*a**3*b**2*d**2*x**4 + 480*sqrt(x)*a**2*b**3*c**2*x**3 - 2688*sqrt(x) 
*a**2*b**3*c*d*x**4 + 560*sqrt(x)*a**2*b**3*d**2*x**5 + 1920*sqrt(x)*a*b** 
4*c**2*x**4 - 1792*sqrt(x)*a*b**4*c*d*x**5 + 1280*sqrt(x)*b**5*c**2*x**5)) 
/(105*sqrt(a + b*x)*a**6*x**4*(a + b*x))