\(\int \frac {(a+b x^2)^2}{x^5 (c+d x)^{5/2}} \, dx\) [562]

Optimal result

Integrand size = 22, antiderivative size = 231 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)^{5/2}} \, dx=\frac {2 \left (b c^2+a d^2\right )^2}{3 c^5 (c+d x)^{3/2}}+\frac {2 \left (b c^2+a d^2\right ) \left (b c^2+5 a d^2\right )}{c^6 \sqrt {c+d x}}-\frac {a^2 \sqrt {c+d x}}{4 c^3 x^4}+\frac {23 a^2 d \sqrt {c+d x}}{24 c^4 x^3}-\frac {a \left (96 b c^2+259 a d^2\right ) \sqrt {c+d x}}{96 c^5 x^2}+\frac {a d \left (352 b c^2+515 a d^2\right ) \sqrt {c+d x}}{64 c^6 x}-\frac {\left (128 b^2 c^4+1120 a b c^2 d^2+1155 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{64 c^{13/2}} \] Output:

2/3*(a*d^2+b*c^2)^2/c^5/(d*x+c)^(3/2)+2*(a*d^2+b*c^2)*(5*a*d^2+b*c^2)/c^6/ 
(d*x+c)^(1/2)-1/4*a^2*(d*x+c)^(1/2)/c^3/x^4+23/24*a^2*d*(d*x+c)^(1/2)/c^4/ 
x^3-1/96*a*(259*a*d^2+96*b*c^2)*(d*x+c)^(1/2)/c^5/x^2+1/64*a*d*(515*a*d^2+ 
352*b*c^2)*(d*x+c)^(1/2)/c^6/x-1/64*(1155*a^2*d^4+1120*a*b*c^2*d^2+128*b^2 
*c^4)*arctanh((d*x+c)^(1/2)/c^(1/2))/c^(13/2)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)^{5/2}} \, dx=\frac {\frac {\sqrt {c} \left (128 b^2 c^4 x^4 (4 c+3 d x)+32 a b c^2 x^2 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )+a^2 \left (-48 c^5+88 c^4 d x-198 c^3 d^2 x^2+693 c^2 d^3 x^3+4620 c d^4 x^4+3465 d^5 x^5\right )\right )}{x^4 (c+d x)^{3/2}}-3 \left (128 b^2 c^4+1120 a b c^2 d^2+1155 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{192 c^{13/2}} \] Input:

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

Output:

((Sqrt[c]*(128*b^2*c^4*x^4*(4*c + 3*d*x) + 32*a*b*c^2*x^2*(-6*c^3 + 21*c^2 
*d*x + 140*c*d^2*x^2 + 105*d^3*x^3) + a^2*(-48*c^5 + 88*c^4*d*x - 198*c^3* 
d^2*x^2 + 693*c^2*d^3*x^3 + 4620*c*d^4*x^4 + 3465*d^5*x^5)))/(x^4*(c + d*x 
)^(3/2)) - 3*(128*b^2*c^4 + 1120*a*b*c^2*d^2 + 1155*a^2*d^4)*ArcTanh[Sqrt[ 
c + d*x]/Sqrt[c]])/(192*c^(13/2))
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {517, 25, 1582, 2336, 25, 1582, 27, 1582, 1584, 2009}

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.

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

\(\Big \downarrow \) 517

\(\displaystyle 2 \int \frac {\left (b c^2-2 b (c+d x) c+a d^2+b (c+d x)^2\right )^2}{d^5 x^5 (c+d x)^2}d\sqrt {c+d x}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int -\frac {\left (b c^2-2 b (c+d x) c+a d^2+b (c+d x)^2\right )^2}{d^5 x^5 (c+d x)^2}d\sqrt {c+d x}\)

\(\Big \downarrow \) 1582

\(\displaystyle 2 \left (-\frac {\int \frac {-8 b^2 c^3 (c+d x)^3+\left (24 b^2 c^4+7 a^2 d^4\right ) (c+d x)^2-8 c \left (3 b c^2-a d^2\right ) \left (b c^2+a d^2\right ) (c+d x)+8 c^2 \left (b c^2+a d^2\right )^2}{d^4 x^4 (c+d x)^2}d\sqrt {c+d x}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

\(\Big \downarrow \) 2336

\(\displaystyle 2 \left (-\frac {-\frac {\int \frac {48 c^2 \left (b c^2+a d^2\right )^2+\left (48 b^2 c^4+115 a^2 d^4\right ) (c+d x)^2-96 c \left (b^2 c^4-a^2 d^4\right ) (c+d x)}{d^3 x^3 (c+d x)^2}d\sqrt {c+d x}}{6 c}-\frac {23 a^2 d \sqrt {c+d x}}{6 c x^3}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (-\frac {\frac {\int -\frac {48 c^2 \left (b c^2+a d^2\right )^2+\left (48 b^2 c^4+115 a^2 d^4\right ) (c+d x)^2-96 c \left (b^2 c^4-a^2 d^4\right ) (c+d x)}{d^3 x^3 (c+d x)^2}d\sqrt {c+d x}}{6 c}-\frac {23 a^2 d \sqrt {c+d x}}{6 c x^3}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

\(\Big \downarrow \) 1582

\(\displaystyle 2 \left (-\frac {\frac {\frac {\int \frac {3 \left (64 \left (b c^2+a d^2\right )^2 c^4-64 \left (b c^2-3 a d^2\right ) \left (b c^2+a d^2\right ) (c+d x) c^3+a d^2 \left (96 b c^2+259 a d^2\right ) (c+d x)^2 c^2\right )}{d^2 x^2 (c+d x)^2}d\sqrt {c+d x}}{4 c^3}+\frac {a \sqrt {c+d x} \left (259 a d^2+96 b c^2\right )}{4 c x^2}}{6 c}-\frac {23 a^2 d \sqrt {c+d x}}{6 c x^3}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (-\frac {\frac {\frac {3 \int \frac {64 \left (b c^2+a d^2\right )^2 c^4-64 \left (b c^2-3 a d^2\right ) \left (b c^2+a d^2\right ) (c+d x) c^3+a d^2 \left (96 b c^2+259 a d^2\right ) (c+d x)^2 c^2}{d^2 x^2 (c+d x)^2}d\sqrt {c+d x}}{4 c^3}+\frac {a \sqrt {c+d x} \left (259 a d^2+96 b c^2\right )}{4 c x^2}}{6 c}-\frac {23 a^2 d \sqrt {c+d x}}{6 c x^3}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

\(\Big \downarrow \) 1582

\(\displaystyle 2 \left (-\frac {\frac {\frac {3 \left (\frac {\int -\frac {128 \left (b c^2+a d^2\right )^2 c^6+512 a d^2 \left (b c^2+a d^2\right ) (c+d x) c^5+a d^2 \left (352 b c^2+515 a d^2\right ) (c+d x)^2 c^4}{d x (c+d x)^2}d\sqrt {c+d x}}{2 c^3}-\frac {a c d \sqrt {c+d x} \left (515 a d^2+352 b c^2\right )}{2 x}\right )}{4 c^3}+\frac {a \sqrt {c+d x} \left (259 a d^2+96 b c^2\right )}{4 c x^2}}{6 c}-\frac {23 a^2 d \sqrt {c+d x}}{6 c x^3}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

\(\Big \downarrow \) 1584

\(\displaystyle 2 \left (-\frac {\frac {\frac {3 \left (\frac {\int \left (\frac {128 \left (b c^2+a d^2\right )^2 c^5}{(c+d x)^2}+\frac {128 \left (b c^2+a d^2\right ) \left (b c^2+5 a d^2\right ) c^4}{c+d x}-\frac {128 b^2 c^8+1120 a b d^2 c^6+1155 a^2 d^4 c^4}{d x}\right )d\sqrt {c+d x}}{2 c^3}-\frac {a c d \sqrt {c+d x} \left (515 a d^2+352 b c^2\right )}{2 x}\right )}{4 c^3}+\frac {a \sqrt {c+d x} \left (259 a d^2+96 b c^2\right )}{4 c x^2}}{6 c}-\frac {23 a^2 d \sqrt {c+d x}}{6 c x^3}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {\frac {\frac {3 \left (\frac {c^{7/2} \left (1155 a^2 d^4+1120 a b c^2 d^2+128 b^2 c^4\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )-\frac {128 c^5 \left (a d^2+b c^2\right )^2}{3 (c+d x)^{3/2}}-\frac {128 c^4 \left (a d^2+b c^2\right ) \left (5 a d^2+b c^2\right )}{\sqrt {c+d x}}}{2 c^3}-\frac {a c d \sqrt {c+d x} \left (515 a d^2+352 b c^2\right )}{2 x}\right )}{4 c^3}+\frac {a \sqrt {c+d x} \left (259 a d^2+96 b c^2\right )}{4 c x^2}}{6 c}-\frac {23 a^2 d \sqrt {c+d x}}{6 c x^3}}{8 c^3}-\frac {a^2 \sqrt {c+d x}}{8 c^3 x^4}\right )\)

Input:

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

Output:

2*(-1/8*(a^2*Sqrt[c + d*x])/(c^3*x^4) - ((-23*a^2*d*Sqrt[c + d*x])/(6*c*x^ 
3) + ((a*(96*b*c^2 + 259*a*d^2)*Sqrt[c + d*x])/(4*c*x^2) + (3*(-1/2*(a*c*d 
*(352*b*c^2 + 515*a*d^2)*Sqrt[c + d*x])/x + ((-128*c^5*(b*c^2 + a*d^2)^2)/ 
(3*(c + d*x)^(3/2)) - (128*c^4*(b*c^2 + a*d^2)*(b*c^2 + 5*a*d^2))/Sqrt[c + 
 d*x] + c^(7/2)*(128*b^2*c^4 + 1120*a*b*c^2*d^2 + 1155*a^2*d^4)*ArcTanh[Sq 
rt[c + d*x]/Sqrt[c]])/(2*c^3)))/(4*c^3))/(6*c))/(8*c^3))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 517

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> Simp[2*(e^m/d^(m + 2*p + 1))   Subst[Int[x^(2*n + 1)*(-c + x^ 
2)^m*(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4)^p, x], x, Sqrt[c + d*x]], x] /; Fr 
eeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[m, 0] && IntegerQ[n + 1/2]
 

rule 1582

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 
4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d 
+ e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ 
(2*p)*(q + 1))   Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e 
*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - 
 b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] 
&& ILtQ[m/2, 0]
 

rule 1584

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ 
b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2336

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema 
inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) 
^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* 
b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(-\frac {33 \left (\frac {35 \left (d x +c \right )^{\frac {3}{2}} x^{4} \left (a^{2} d^{4}+\frac {32}{33} b \,c^{2} d^{2} a +\frac {128}{1155} b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2}+\frac {8 \left (-\frac {32}{3} b^{2} x^{4}+4 a b \,x^{2}+a^{2}\right ) c^{\frac {11}{2}}}{33}+d x \left (\frac {4 \left (-\frac {16}{11} b^{2} x^{4}-\frac {28}{11} a b \,x^{2}-\frac {1}{3} a^{2}\right ) c^{\frac {9}{2}}}{3}+d x a \left (-\frac {7 d x \left (\frac {160 b \,x^{2}}{33}+a \right ) c^{\frac {5}{2}}}{2}+\left (-\frac {2240 b \,x^{2}}{99}+a \right ) c^{\frac {7}{2}}-\frac {35 d^{2} \left (d x \sqrt {c}+\frac {4 c^{\frac {3}{2}}}{3}\right ) x^{2} a}{2}\right )\right )\right )}{32 c^{\frac {13}{2}} \left (d x +c \right )^{\frac {3}{2}} x^{4}}\)	\(174\)
risch	\(-\frac {\sqrt {d x +c}\, a \left (-1545 a \,x^{3} d^{3}-1056 b \,c^{2} d \,x^{3}+518 a \,d^{2} x^{2} c +192 b \,c^{3} x^{2}-184 a d x \,c^{2}+48 c^{3} a \right )}{192 c^{6} x^{4}}+\frac {-\frac {2 \left (-640 a^{2} d^{4}-768 b \,c^{2} d^{2} a -128 b^{2} c^{4}\right )}{\sqrt {d x +c}}+\frac {256 c \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (1155 a^{2} d^{4}+1120 b \,c^{2} d^{2} a +128 b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}}{128 c^{6}}\)	\(192\)
derivativedivides	\(-\frac {2 \left (\frac {\left (-\frac {515}{128} a^{2} d^{4}-\frac {11}{4} b \,c^{2} d^{2} a \right ) \left (d x +c \right )^{\frac {7}{2}}+\left (\frac {5153}{384} a^{2} c \,d^{4}+\frac {35}{4} a \,c^{3} d^{2} b \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5855}{384} a^{2} c^{2} d^{4}-\frac {37}{4} a b \,c^{4} d^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {765}{128} a^{2} c^{3} d^{4}+\frac {13}{4} a b \,c^{5} d^{2}\right ) \sqrt {d x +c}}{d^{4} x^{4}}+\frac {\left (1155 a^{2} d^{4}+1120 b \,c^{2} d^{2} a +128 b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}\right )}{c^{6}}-\frac {2 \left (-5 a^{2} d^{4}-6 b \,c^{2} d^{2} a -b^{2} c^{4}\right )}{c^{6} \sqrt {d x +c}}-\frac {2 \left (-a^{2} d^{4}-2 b \,c^{2} d^{2} a -b^{2} c^{4}\right )}{3 c^{5} \left (d x +c \right )^{\frac {3}{2}}}\)	\(253\)
default	\(-\frac {2 \left (\frac {\left (-\frac {515}{128} a^{2} d^{4}-\frac {11}{4} b \,c^{2} d^{2} a \right ) \left (d x +c \right )^{\frac {7}{2}}+\left (\frac {5153}{384} a^{2} c \,d^{4}+\frac {35}{4} a \,c^{3} d^{2} b \right ) \left (d x +c \right )^{\frac {5}{2}}+\left (-\frac {5855}{384} a^{2} c^{2} d^{4}-\frac {37}{4} a b \,c^{4} d^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {765}{128} a^{2} c^{3} d^{4}+\frac {13}{4} a b \,c^{5} d^{2}\right ) \sqrt {d x +c}}{d^{4} x^{4}}+\frac {\left (1155 a^{2} d^{4}+1120 b \,c^{2} d^{2} a +128 b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}\right )}{c^{6}}-\frac {2 \left (-5 a^{2} d^{4}-6 b \,c^{2} d^{2} a -b^{2} c^{4}\right )}{c^{6} \sqrt {d x +c}}-\frac {2 \left (-a^{2} d^{4}-2 b \,c^{2} d^{2} a -b^{2} c^{4}\right )}{3 c^{5} \left (d x +c \right )^{\frac {3}{2}}}\)	\(253\)

Input:

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

Output:

-33/32*(35/2*(d*x+c)^(3/2)*x^4*(a^2*d^4+32/33*b*c^2*d^2*a+128/1155*b^2*c^4 
)*arctanh((d*x+c)^(1/2)/c^(1/2))+8/33*(-32/3*b^2*x^4+4*a*b*x^2+a^2)*c^(11/ 
2)+d*x*(4/3*(-16/11*b^2*x^4-28/11*a*b*x^2-1/3*a^2)*c^(9/2)+d*x*a*(-7/2*d*x 
*(160/33*b*x^2+a)*c^(5/2)+(-2240/99*b*x^2+a)*c^(7/2)-35/2*d^2*(d*x*c^(1/2) 
+4/3*c^(3/2))*x^2*a)))/c^(13/2)/(d*x+c)^(3/2)/x^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.65 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left ({\left (128 \, b^{2} c^{4} d^{2} + 1120 \, a b c^{2} d^{4} + 1155 \, a^{2} d^{6}\right )} x^{6} + 2 \, {\left (128 \, b^{2} c^{5} d + 1120 \, a b c^{3} d^{3} + 1155 \, a^{2} c d^{5}\right )} x^{5} + {\left (128 \, b^{2} c^{6} + 1120 \, a b c^{4} d^{2} + 1155 \, a^{2} c^{2} d^{4}\right )} x^{4}\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (88 \, a^{2} c^{5} d x - 48 \, a^{2} c^{6} + 3 \, {\left (128 \, b^{2} c^{5} d + 1120 \, a b c^{3} d^{3} + 1155 \, a^{2} c d^{5}\right )} x^{5} + 4 \, {\left (128 \, b^{2} c^{6} + 1120 \, a b c^{4} d^{2} + 1155 \, a^{2} c^{2} d^{4}\right )} x^{4} + 21 \, {\left (32 \, a b c^{5} d + 33 \, a^{2} c^{3} d^{3}\right )} x^{3} - 6 \, {\left (32 \, a b c^{6} + 33 \, a^{2} c^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x + c}}{384 \, {\left (c^{7} d^{2} x^{6} + 2 \, c^{8} d x^{5} + c^{9} x^{4}\right )}}, \frac {3 \, {\left ({\left (128 \, b^{2} c^{4} d^{2} + 1120 \, a b c^{2} d^{4} + 1155 \, a^{2} d^{6}\right )} x^{6} + 2 \, {\left (128 \, b^{2} c^{5} d + 1120 \, a b c^{3} d^{3} + 1155 \, a^{2} c d^{5}\right )} x^{5} + {\left (128 \, b^{2} c^{6} + 1120 \, a b c^{4} d^{2} + 1155 \, a^{2} c^{2} d^{4}\right )} x^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) + {\left (88 \, a^{2} c^{5} d x - 48 \, a^{2} c^{6} + 3 \, {\left (128 \, b^{2} c^{5} d + 1120 \, a b c^{3} d^{3} + 1155 \, a^{2} c d^{5}\right )} x^{5} + 4 \, {\left (128 \, b^{2} c^{6} + 1120 \, a b c^{4} d^{2} + 1155 \, a^{2} c^{2} d^{4}\right )} x^{4} + 21 \, {\left (32 \, a b c^{5} d + 33 \, a^{2} c^{3} d^{3}\right )} x^{3} - 6 \, {\left (32 \, a b c^{6} + 33 \, a^{2} c^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x + c}}{192 \, {\left (c^{7} d^{2} x^{6} + 2 \, c^{8} d x^{5} + c^{9} x^{4}\right )}}\right ] \] Input:

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

Output:

[1/384*(3*((128*b^2*c^4*d^2 + 1120*a*b*c^2*d^4 + 1155*a^2*d^6)*x^6 + 2*(12 
8*b^2*c^5*d + 1120*a*b*c^3*d^3 + 1155*a^2*c*d^5)*x^5 + (128*b^2*c^6 + 1120 
*a*b*c^4*d^2 + 1155*a^2*c^2*d^4)*x^4)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*s 
qrt(c) + 2*c)/x) + 2*(88*a^2*c^5*d*x - 48*a^2*c^6 + 3*(128*b^2*c^5*d + 112 
0*a*b*c^3*d^3 + 1155*a^2*c*d^5)*x^5 + 4*(128*b^2*c^6 + 1120*a*b*c^4*d^2 + 
1155*a^2*c^2*d^4)*x^4 + 21*(32*a*b*c^5*d + 33*a^2*c^3*d^3)*x^3 - 6*(32*a*b 
*c^6 + 33*a^2*c^4*d^2)*x^2)*sqrt(d*x + c))/(c^7*d^2*x^6 + 2*c^8*d*x^5 + c^ 
9*x^4), 1/192*(3*((128*b^2*c^4*d^2 + 1120*a*b*c^2*d^4 + 1155*a^2*d^6)*x^6 
+ 2*(128*b^2*c^5*d + 1120*a*b*c^3*d^3 + 1155*a^2*c*d^5)*x^5 + (128*b^2*c^6 
 + 1120*a*b*c^4*d^2 + 1155*a^2*c^2*d^4)*x^4)*sqrt(-c)*arctan(sqrt(-c)/sqrt 
(d*x + c)) + (88*a^2*c^5*d*x - 48*a^2*c^6 + 3*(128*b^2*c^5*d + 1120*a*b*c^ 
3*d^3 + 1155*a^2*c*d^5)*x^5 + 4*(128*b^2*c^6 + 1120*a*b*c^4*d^2 + 1155*a^2 
*c^2*d^4)*x^4 + 21*(32*a*b*c^5*d + 33*a^2*c^3*d^3)*x^3 - 6*(32*a*b*c^6 + 3 
3*a^2*c^4*d^2)*x^2)*sqrt(d*x + c))/(c^7*d^2*x^6 + 2*c^8*d*x^5 + c^9*x^4)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.58 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)^{5/2}} \, dx=\frac {1}{384} \, d^{4} {\left (\frac {2 \, {\left (128 \, b^{2} c^{9} + 256 \, a b c^{7} d^{2} + 128 \, a^{2} c^{5} d^{4} + 3 \, {\left (128 \, b^{2} c^{4} + 1120 \, a b c^{2} d^{2} + 1155 \, a^{2} d^{4}\right )} {\left (d x + c\right )}^{5} - 11 \, {\left (128 \, b^{2} c^{5} + 1120 \, a b c^{3} d^{2} + 1155 \, a^{2} c d^{4}\right )} {\left (d x + c\right )}^{4} + 7 \, {\left (256 \, b^{2} c^{6} + 2336 \, a b c^{4} d^{2} + 2409 \, a^{2} c^{2} d^{4}\right )} {\left (d x + c\right )}^{3} - 3 \, {\left (256 \, b^{2} c^{7} + 2976 \, a b c^{5} d^{2} + 3069 \, a^{2} c^{3} d^{4}\right )} {\left (d x + c\right )}^{2} - 128 \, {\left (b^{2} c^{8} - 10 \, a b c^{6} d^{2} - 11 \, a^{2} c^{4} d^{4}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {11}{2}} c^{6} d^{4} - 4 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{7} d^{4} + 6 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{8} d^{4} - 4 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{9} d^{4} + {\left (d x + c\right )}^{\frac {3}{2}} c^{10} d^{4}} + \frac {3 \, {\left (128 \, b^{2} c^{4} + 1120 \, a b c^{2} d^{2} + 1155 \, a^{2} d^{4}\right )} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{c^{\frac {13}{2}} d^{4}}\right )} \] Input:

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

Output:

1/384*d^4*(2*(128*b^2*c^9 + 256*a*b*c^7*d^2 + 128*a^2*c^5*d^4 + 3*(128*b^2 
*c^4 + 1120*a*b*c^2*d^2 + 1155*a^2*d^4)*(d*x + c)^5 - 11*(128*b^2*c^5 + 11 
20*a*b*c^3*d^2 + 1155*a^2*c*d^4)*(d*x + c)^4 + 7*(256*b^2*c^6 + 2336*a*b*c 
^4*d^2 + 2409*a^2*c^2*d^4)*(d*x + c)^3 - 3*(256*b^2*c^7 + 2976*a*b*c^5*d^2 
 + 3069*a^2*c^3*d^4)*(d*x + c)^2 - 128*(b^2*c^8 - 10*a*b*c^6*d^2 - 11*a^2* 
c^4*d^4)*(d*x + c))/((d*x + c)^(11/2)*c^6*d^4 - 4*(d*x + c)^(9/2)*c^7*d^4 
+ 6*(d*x + c)^(7/2)*c^8*d^4 - 4*(d*x + c)^(5/2)*c^9*d^4 + (d*x + c)^(3/2)* 
c^10*d^4) + 3*(128*b^2*c^4 + 1120*a*b*c^2*d^2 + 1155*a^2*d^4)*log((sqrt(d* 
x + c) - sqrt(c))/(sqrt(d*x + c) + sqrt(c)))/(c^(13/2)*d^4))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)^{5/2}} \, dx=\frac {{\left (128 \, b^{2} c^{4} + 1120 \, a b c^{2} d^{2} + 1155 \, a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{64 \, \sqrt {-c} c^{6}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )} b^{2} c^{4} + b^{2} c^{5} + 18 \, {\left (d x + c\right )} a b c^{2} d^{2} + 2 \, a b c^{3} d^{2} + 15 \, {\left (d x + c\right )} a^{2} d^{4} + a^{2} c d^{4}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{6}} + \frac {1056 \, {\left (d x + c\right )}^{\frac {7}{2}} a b c^{2} d^{2} - 3360 \, {\left (d x + c\right )}^{\frac {5}{2}} a b c^{3} d^{2} + 3552 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c^{4} d^{2} - 1248 \, \sqrt {d x + c} a b c^{5} d^{2} + 1545 \, {\left (d x + c\right )}^{\frac {7}{2}} a^{2} d^{4} - 5153 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} c d^{4} + 5855 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} c^{2} d^{4} - 2295 \, \sqrt {d x + c} a^{2} c^{3} d^{4}}{192 \, c^{6} d^{4} x^{4}} \] Input:

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

Output:

1/64*(128*b^2*c^4 + 1120*a*b*c^2*d^2 + 1155*a^2*d^4)*arctan(sqrt(d*x + c)/ 
sqrt(-c))/(sqrt(-c)*c^6) + 2/3*(3*(d*x + c)*b^2*c^4 + b^2*c^5 + 18*(d*x + 
c)*a*b*c^2*d^2 + 2*a*b*c^3*d^2 + 15*(d*x + c)*a^2*d^4 + a^2*c*d^4)/((d*x + 
 c)^(3/2)*c^6) + 1/192*(1056*(d*x + c)^(7/2)*a*b*c^2*d^2 - 3360*(d*x + c)^ 
(5/2)*a*b*c^3*d^2 + 3552*(d*x + c)^(3/2)*a*b*c^4*d^2 - 1248*sqrt(d*x + c)* 
a*b*c^5*d^2 + 1545*(d*x + c)^(7/2)*a^2*d^4 - 5153*(d*x + c)^(5/2)*a^2*c*d^ 
4 + 5855*(d*x + c)^(3/2)*a^2*c^2*d^4 - 2295*sqrt(d*x + c)*a^2*c^3*d^4)/(c^ 
6*d^4*x^4)
 

Mupad [B] (verification not implemented)

Time = 9.64 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)^{5/2}} \, dx=\frac {\frac {2\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}{3\,c}-\frac {11\,{\left (c+d\,x\right )}^4\,\left (1155\,a^2\,d^4+1120\,a\,b\,c^2\,d^2+128\,b^2\,c^4\right )}{192\,c^5}+\frac {{\left (c+d\,x\right )}^5\,\left (1155\,a^2\,d^4+1120\,a\,b\,c^2\,d^2+128\,b^2\,c^4\right )}{64\,c^6}+\frac {7\,{\left (c+d\,x\right )}^3\,\left (2409\,a^2\,d^4+2336\,a\,b\,c^2\,d^2+256\,b^2\,c^4\right )}{192\,c^4}-\frac {{\left (c+d\,x\right )}^2\,\left (3069\,a^2\,d^4+2976\,a\,b\,c^2\,d^2+256\,b^2\,c^4\right )}{64\,c^3}+\frac {2\,\left (c+d\,x\right )\,\left (11\,a^2\,d^4+10\,a\,b\,c^2\,d^2-b^2\,c^4\right )}{3\,c^2}}{{\left (c+d\,x\right )}^{11/2}-4\,c\,{\left (c+d\,x\right )}^{9/2}+c^4\,{\left (c+d\,x\right )}^{3/2}-4\,c^3\,{\left (c+d\,x\right )}^{5/2}+6\,c^2\,{\left (c+d\,x\right )}^{7/2}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c}}\right )\,\left (1155\,a^2\,d^4+1120\,a\,b\,c^2\,d^2+128\,b^2\,c^4\right )}{64\,c^{13/2}} \] Input:

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

Output:

((2*(a^2*d^4 + b^2*c^4 + 2*a*b*c^2*d^2))/(3*c) - (11*(c + d*x)^4*(1155*a^2 
*d^4 + 128*b^2*c^4 + 1120*a*b*c^2*d^2))/(192*c^5) + ((c + d*x)^5*(1155*a^2 
*d^4 + 128*b^2*c^4 + 1120*a*b*c^2*d^2))/(64*c^6) + (7*(c + d*x)^3*(2409*a^ 
2*d^4 + 256*b^2*c^4 + 2336*a*b*c^2*d^2))/(192*c^4) - ((c + d*x)^2*(3069*a^ 
2*d^4 + 256*b^2*c^4 + 2976*a*b*c^2*d^2))/(64*c^3) + (2*(c + d*x)*(11*a^2*d 
^4 - b^2*c^4 + 10*a*b*c^2*d^2))/(3*c^2))/((c + d*x)^(11/2) - 4*c*(c + d*x) 
^(9/2) + c^4*(c + d*x)^(3/2) - 4*c^3*(c + d*x)^(5/2) + 6*c^2*(c + d*x)^(7/ 
2)) - (atanh((c + d*x)^(1/2)/c^(1/2))*(1155*a^2*d^4 + 128*b^2*c^4 + 1120*a 
*b*c^2*d^2))/(64*c^(13/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.33 \[ \int \frac {\left (a+b x^2\right )^2}{x^5 (c+d x)^{5/2}} \, dx=\frac {3465 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{2} c \,d^{4} x^{4}+3465 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{2} d^{5} x^{5}+3360 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a b \,c^{3} d^{2} x^{4}+3360 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a b \,c^{2} d^{3} x^{5}+384 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b^{2} c^{5} x^{4}+384 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b^{2} c^{4} d \,x^{5}-3465 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{2} c \,d^{4} x^{4}-3465 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{2} d^{5} x^{5}-3360 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a b \,c^{3} d^{2} x^{4}-3360 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a b \,c^{2} d^{3} x^{5}-384 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b^{2} c^{5} x^{4}-384 \sqrt {c}\, \sqrt {d x +c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b^{2} c^{4} d \,x^{5}-96 a^{2} c^{6}+176 a^{2} c^{5} d x -396 a^{2} c^{4} d^{2} x^{2}+1386 a^{2} c^{3} d^{3} x^{3}+9240 a^{2} c^{2} d^{4} x^{4}+6930 a^{2} c \,d^{5} x^{5}-384 a b \,c^{6} x^{2}+1344 a b \,c^{5} d \,x^{3}+8960 a b \,c^{4} d^{2} x^{4}+6720 a b \,c^{3} d^{3} x^{5}+1024 b^{2} c^{6} x^{4}+768 b^{2} c^{5} d \,x^{5}}{384 \sqrt {d x +c}\, c^{7} x^{4} \left (d x +c \right )} \] Input:

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

Output:

(3465*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*a**2*c*d**4*x**4 
+ 3465*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*a**2*d**5*x**5 + 
 3360*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*a*b*c**3*d**2*x** 
4 + 3360*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*a*b*c**2*d**3* 
x**5 + 384*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*b**2*c**5*x* 
*4 + 384*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) - sqrt(c))*b**2*c**4*d*x* 
*5 - 3465*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*a**2*c*d**4*x 
**4 - 3465*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*a**2*d**5*x* 
*5 - 3360*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*a*b*c**3*d**2 
*x**4 - 3360*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*a*b*c**2*d 
**3*x**5 - 384*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*b**2*c** 
5*x**4 - 384*sqrt(c)*sqrt(c + d*x)*log(sqrt(c + d*x) + sqrt(c))*b**2*c**4* 
d*x**5 - 96*a**2*c**6 + 176*a**2*c**5*d*x - 396*a**2*c**4*d**2*x**2 + 1386 
*a**2*c**3*d**3*x**3 + 9240*a**2*c**2*d**4*x**4 + 6930*a**2*c*d**5*x**5 - 
384*a*b*c**6*x**2 + 1344*a*b*c**5*d*x**3 + 8960*a*b*c**4*d**2*x**4 + 6720* 
a*b*c**3*d**3*x**5 + 1024*b**2*c**6*x**4 + 768*b**2*c**5*d*x**5)/(384*sqrt 
(c + d*x)*c**7*x**4*(c + d*x))