\(\int \frac {(c+d x)^{5/2} (a+b x^2)^2}{(e x)^{15/2}} \, dx\) [820]

Optimal result

Integrand size = 26, antiderivative size = 270 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=-\frac {2 b^2 c^2 \sqrt {c+d x}}{5 e^5 (e x)^{5/2}}-\frac {22 b^2 c d \sqrt {c+d x}}{15 e^6 (e x)^{3/2}}-\frac {46 b^2 d^2 \sqrt {c+d x}}{15 e^7 \sqrt {e x}}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}+\frac {12 a^2 d (c+d x)^{7/2}}{143 c^2 e^2 (e x)^{11/2}}-\frac {4 a \left (143 b c^2+12 a d^2\right ) (c+d x)^{7/2}}{1287 c^3 e^3 (e x)^{9/2}}+\frac {8 a d \left (143 b c^2+12 a d^2\right ) (c+d x)^{7/2}}{9009 c^4 e^4 (e x)^{7/2}}+\frac {2 b^2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{e^{15/2}} \] Output:

-2/5*b^2*c^2*(d*x+c)^(1/2)/e^5/(e*x)^(5/2)-22/15*b^2*c*d*(d*x+c)^(1/2)/e^6 
/(e*x)^(3/2)-46/15*b^2*d^2*(d*x+c)^(1/2)/e^7/(e*x)^(1/2)-2/13*a^2*(d*x+c)^ 
(7/2)/c/e/(e*x)^(13/2)+12/143*a^2*d*(d*x+c)^(7/2)/c^2/e^2/(e*x)^(11/2)-4/1 
287*a*(12*a*d^2+143*b*c^2)*(d*x+c)^(7/2)/c^3/e^3/(e*x)^(9/2)+8/9009*a*d*(1 
2*a*d^2+143*b*c^2)*(d*x+c)^(7/2)/c^4/e^4/(e*x)^(7/2)+2*b^2*d^(5/2)*arctanh 
(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/e^(15/2)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.63 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=-\frac {2 \sqrt {e x} \left (\sqrt {c+d x} \left (1430 a b c^2 x^2 (7 c-2 d x) (c+d x)^3+3003 b^2 c^4 x^4 \left (3 c^2+11 c d x+23 d^2 x^2\right )+15 a^2 (c+d x)^3 \left (231 c^3-126 c^2 d x+56 c d^2 x^2-16 d^3 x^3\right )\right )+45045 b^2 c^4 d^{5/2} x^{13/2} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )\right )}{45045 c^4 e^8 x^7} \] Input:

Integrate[((c + d*x)^(5/2)*(a + b*x^2)^2)/(e*x)^(15/2),x]
 

Output:

(-2*Sqrt[e*x]*(Sqrt[c + d*x]*(1430*a*b*c^2*x^2*(7*c - 2*d*x)*(c + d*x)^3 + 
 3003*b^2*c^4*x^4*(3*c^2 + 11*c*d*x + 23*d^2*x^2) + 15*a^2*(c + d*x)^3*(23 
1*c^3 - 126*c^2*d*x + 56*c*d^2*x^2 - 16*d^3*x^3)) + 45045*b^2*c^4*d^(5/2)* 
x^(13/2)*Log[-(Sqrt[d]*Sqrt[x]) + Sqrt[c + d*x]]))/(45045*c^4*e^8*x^7)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {520, 27, 2124, 27, 520, 27, 87, 57, 57, 57, 65, 221}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^{5/2}}{(e x)^{15/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {(c+d x)^{5/2} \left (-13 b^2 c x^3-26 a b c x+6 a^2 d\right )}{2 (e x)^{13/2}}dx}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(c+d x)^{5/2} \left (-13 b^2 c x^3-26 a b c x+6 a^2 d\right )}{(e x)^{13/2}}dx}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {(c+d x)^{5/2} \left (143 b^2 c^2 x^2+2 a \left (143 b c^2+12 a d^2\right )\right )}{2 (e x)^{11/2}}dx}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^{5/2} \left (143 b^2 c^2 x^2+2 a \left (143 b c^2+12 a d^2\right )\right )}{(e x)^{11/2}}dx}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {\left (4 a d \left (143 b c^2+12 a d^2\right )-1287 b^2 c^3 x\right ) (c+d x)^{5/2}}{2 (e x)^{9/2}}dx}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (4 a d \left (143 b c^2+12 a d^2\right )-1287 b^2 c^3 x\right ) (c+d x)^{5/2}}{(e x)^{9/2}}dx}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {1287 b^2 c^3 \int \frac {(c+d x)^{5/2}}{(e x)^{7/2}}dx}{e}-\frac {8 a d (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 57

\(\displaystyle -\frac {-\frac {-\frac {-\frac {1287 b^2 c^3 \left (\frac {d \int \frac {(c+d x)^{3/2}}{(e x)^{5/2}}dx}{e}-\frac {2 (c+d x)^{5/2}}{5 e (e x)^{5/2}}\right )}{e}-\frac {8 a d (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 57

\(\displaystyle -\frac {-\frac {-\frac {-\frac {1287 b^2 c^3 \left (\frac {d \left (\frac {d \int \frac {\sqrt {c+d x}}{(e x)^{3/2}}dx}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {2 (c+d x)^{5/2}}{5 e (e x)^{5/2}}\right )}{e}-\frac {8 a d (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 57

\(\displaystyle -\frac {-\frac {-\frac {-\frac {1287 b^2 c^3 \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{e}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {2 (c+d x)^{5/2}}{5 e (e x)^{5/2}}\right )}{e}-\frac {8 a d (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 65

\(\displaystyle -\frac {-\frac {-\frac {-\frac {1287 b^2 c^3 \left (\frac {d \left (\frac {d \left (\frac {2 d \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{e}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {2 (c+d x)^{5/2}}{5 e (e x)^{5/2}}\right )}{e}-\frac {8 a d (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {12 a^2 d (c+d x)^{7/2}}{11 c e (e x)^{11/2}}-\frac {-\frac {-\frac {8 a d (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}-\frac {1287 b^2 c^3 \left (\frac {d \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{e^{3/2}}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {2 (c+d x)^{5/2}}{5 e (e x)^{5/2}}\right )}{e}}{9 c e}-\frac {4 a (c+d x)^{7/2} \left (12 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}}{13 c e}-\frac {2 a^2 (c+d x)^{7/2}}{13 c e (e x)^{13/2}}\)

Input:

Int[((c + d*x)^(5/2)*(a + b*x^2)^2)/(e*x)^(15/2),x]
 

Output:

(-2*a^2*(c + d*x)^(7/2))/(13*c*e*(e*x)^(13/2)) - ((-12*a^2*d*(c + d*x)^(7/ 
2))/(11*c*e*(e*x)^(11/2)) - ((-4*a*(143*b*c^2 + 12*a*d^2)*(c + d*x)^(7/2)) 
/(9*c*e*(e*x)^(9/2)) - ((-8*a*d*(143*b*c^2 + 12*a*d^2)*(c + d*x)^(7/2))/(7 
*c*e*(e*x)^(7/2)) - (1287*b^2*c^3*((-2*(c + d*x)^(5/2))/(5*e*(e*x)^(5/2)) 
+ (d*((-2*(c + d*x)^(3/2))/(3*e*(e*x)^(3/2)) + (d*((-2*Sqrt[c + d*x])/(e*S 
qrt[e*x]) + (2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x]) 
])/e^(3/2)))/e))/e))/e)/(9*c*e))/(11*c*e))/(13*c*e)
 

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 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(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] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 221

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]
 

rule 520

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, e*x, x], R = Pol 
ynomialRemainder[(a + b*x^2)^p, e*x, x]}, Simp[R*(e*x)^(m + 1)*((c + d*x)^( 
n + 1)/((m + 1)*(e*c))), x] + Simp[1/((m + 1)*(e*c))   Int[(e*x)^(m + 1)*(c 
 + d*x)^n*ExpandToSum[(m + 1)*(e*c)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[m, -1] &&  !IntegerQ[n]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.01

method	result	size
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-240 a^{2} d^{6} x^{6}-2860 a b \,c^{2} d^{4} x^{6}+69069 b^{2} c^{4} d^{2} x^{6}+120 a^{2} c \,d^{5} x^{5}+1430 a b \,c^{3} d^{3} x^{5}+33033 b^{2} c^{5} d \,x^{5}-90 a^{2} c^{2} d^{4} x^{4}+21450 a b \,c^{4} d^{2} x^{4}+9009 b^{2} c^{6} x^{4}+75 a^{2} c^{3} d^{3} x^{3}+27170 a b \,c^{5} d \,x^{3}+5565 a^{2} c^{4} d^{2} x^{2}+10010 a b \,c^{6} x^{2}+8505 a^{2} c^{5} d x +3465 a^{2} c^{6}\right )}{45045 x^{6} c^{4} e^{7} \sqrt {e x}}+\frac {b^{2} d^{3} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{\sqrt {d e}\, e^{7} \sqrt {e x}\, \sqrt {d x +c}}\)	\(273\)
default	\(\frac {\sqrt {d x +c}\, \left (45045 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} d^{3} e \,x^{7}+480 a^{2} d^{6} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+5720 a b \,c^{2} d^{4} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-138138 b^{2} c^{4} d^{2} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-240 a^{2} c \,d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-2860 a b \,c^{3} d^{3} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-66066 b^{2} c^{5} d \,x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+180 a^{2} c^{2} d^{4} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-42900 a b \,c^{4} d^{2} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-18018 b^{2} c^{6} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-150 a^{2} c^{3} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-54340 a b \,c^{5} d \,x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-11130 a^{2} c^{4} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-20020 a b \,c^{6} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-17010 a^{2} c^{5} d x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-6930 a^{2} c^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\right )}{45045 e^{7} x^{6} c^{4} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(494\)

Input:

int((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(15/2),x,method=_RETURNVERBOSE)
 

Output:

-2/45045*(d*x+c)^(1/2)*(-240*a^2*d^6*x^6-2860*a*b*c^2*d^4*x^6+69069*b^2*c^ 
4*d^2*x^6+120*a^2*c*d^5*x^5+1430*a*b*c^3*d^3*x^5+33033*b^2*c^5*d*x^5-90*a^ 
2*c^2*d^4*x^4+21450*a*b*c^4*d^2*x^4+9009*b^2*c^6*x^4+75*a^2*c^3*d^3*x^3+27 
170*a*b*c^5*d*x^3+5565*a^2*c^4*d^2*x^2+10010*a*b*c^6*x^2+8505*a^2*c^5*d*x+ 
3465*a^2*c^6)/x^6/c^4/e^7/(e*x)^(1/2)+b^2*d^3*ln((1/2*c*e+d*e*x)/(d*e)^(1/ 
2)+(d*e*x^2+c*e*x)^(1/2))/(d*e)^(1/2)/e^7*((d*x+c)*e*x)^(1/2)/(e*x)^(1/2)/ 
(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\left [\frac {45045 \, b^{2} c^{4} d^{2} e x^{7} \sqrt {\frac {d}{e}} \log \left (2 \, d x + 2 \, \sqrt {d x + c} \sqrt {e x} \sqrt {\frac {d}{e}} + c\right ) - 2 \, {\left (8505 \, a^{2} c^{5} d x + 3465 \, a^{2} c^{6} + {\left (69069 \, b^{2} c^{4} d^{2} - 2860 \, a b c^{2} d^{4} - 240 \, a^{2} d^{6}\right )} x^{6} + {\left (33033 \, b^{2} c^{5} d + 1430 \, a b c^{3} d^{3} + 120 \, a^{2} c d^{5}\right )} x^{5} + 3 \, {\left (3003 \, b^{2} c^{6} + 7150 \, a b c^{4} d^{2} - 30 \, a^{2} c^{2} d^{4}\right )} x^{4} + 5 \, {\left (5434 \, a b c^{5} d + 15 \, a^{2} c^{3} d^{3}\right )} x^{3} + 35 \, {\left (286 \, a b c^{6} + 159 \, a^{2} c^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{45045 \, c^{4} e^{8} x^{7}}, -\frac {2 \, {\left (45045 \, b^{2} c^{4} d^{2} e x^{7} \sqrt {-\frac {d}{e}} \arctan \left (\frac {\sqrt {e x} \sqrt {-\frac {d}{e}}}{\sqrt {d x + c}}\right ) + {\left (8505 \, a^{2} c^{5} d x + 3465 \, a^{2} c^{6} + {\left (69069 \, b^{2} c^{4} d^{2} - 2860 \, a b c^{2} d^{4} - 240 \, a^{2} d^{6}\right )} x^{6} + {\left (33033 \, b^{2} c^{5} d + 1430 \, a b c^{3} d^{3} + 120 \, a^{2} c d^{5}\right )} x^{5} + 3 \, {\left (3003 \, b^{2} c^{6} + 7150 \, a b c^{4} d^{2} - 30 \, a^{2} c^{2} d^{4}\right )} x^{4} + 5 \, {\left (5434 \, a b c^{5} d + 15 \, a^{2} c^{3} d^{3}\right )} x^{3} + 35 \, {\left (286 \, a b c^{6} + 159 \, a^{2} c^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}\right )}}{45045 \, c^{4} e^{8} x^{7}}\right ] \] Input:

integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(15/2),x, algorithm="fricas")
 

Output:

[1/45045*(45045*b^2*c^4*d^2*e*x^7*sqrt(d/e)*log(2*d*x + 2*sqrt(d*x + c)*sq 
rt(e*x)*sqrt(d/e) + c) - 2*(8505*a^2*c^5*d*x + 3465*a^2*c^6 + (69069*b^2*c 
^4*d^2 - 2860*a*b*c^2*d^4 - 240*a^2*d^6)*x^6 + (33033*b^2*c^5*d + 1430*a*b 
*c^3*d^3 + 120*a^2*c*d^5)*x^5 + 3*(3003*b^2*c^6 + 7150*a*b*c^4*d^2 - 30*a^ 
2*c^2*d^4)*x^4 + 5*(5434*a*b*c^5*d + 15*a^2*c^3*d^3)*x^3 + 35*(286*a*b*c^6 
 + 159*a^2*c^4*d^2)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(c^4*e^8*x^7), -2/45045* 
(45045*b^2*c^4*d^2*e*x^7*sqrt(-d/e)*arctan(sqrt(e*x)*sqrt(-d/e)/sqrt(d*x + 
 c)) + (8505*a^2*c^5*d*x + 3465*a^2*c^6 + (69069*b^2*c^4*d^2 - 2860*a*b*c^ 
2*d^4 - 240*a^2*d^6)*x^6 + (33033*b^2*c^5*d + 1430*a*b*c^3*d^3 + 120*a^2*c 
*d^5)*x^5 + 3*(3003*b^2*c^6 + 7150*a*b*c^4*d^2 - 30*a^2*c^2*d^4)*x^4 + 5*( 
5434*a*b*c^5*d + 15*a^2*c^3*d^3)*x^3 + 35*(286*a*b*c^6 + 159*a^2*c^4*d^2)* 
x^2)*sqrt(d*x + c)*sqrt(e*x))/(c^4*e^8*x^7)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(5/2)*(b*x**2+a)**2/(e*x)**(15/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(15/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.31 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=-\frac {2 \, {\left (\frac {45045 \, b^{2} d^{3} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e}} + \frac {{\left (45045 \, b^{2} c^{6} d^{9} e^{6} - {\left (285285 \, b^{2} c^{5} d^{9} e^{6} - {\left (759759 \, b^{2} c^{4} d^{9} e^{6} + {\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {{\left (69069 \, b^{2} c^{6} d^{9} e^{6} - 2860 \, a b c^{4} d^{11} e^{6} - 240 \, a^{2} c^{2} d^{13} e^{6}\right )} {\left (d x + c\right )}}{c^{6}} - \frac {13 \, {\left (29337 \, b^{2} c^{7} d^{9} e^{6} - 1430 \, a b c^{5} d^{11} e^{6} - 120 \, a^{2} c^{3} d^{13} e^{6}\right )}}{c^{6}}\right )} + \frac {143 \, {\left (6153 \, b^{2} c^{8} d^{9} e^{6} - 200 \, a b c^{6} d^{11} e^{6} - 30 \, a^{2} c^{4} d^{13} e^{6}\right )}}{c^{6}}\right )} - \frac {429 \, {\left (2534 \, b^{2} c^{9} d^{9} e^{6} - 30 \, a b c^{7} d^{11} e^{6} - 15 \, a^{2} c^{5} d^{13} e^{6}\right )}}{c^{6}}\right )} {\left (d x + c\right )}\right )} {\left (d x + c\right )}\right )} {\left (d x + c\right )}\right )} \sqrt {d x + c}}{{\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {13}{2}}}\right )} d}{45045 \, e^{7} {\left | d \right |}} \] Input:

integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(15/2),x, algorithm="giac")
 

Output:

-2/45045*(45045*b^2*d^3*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)* 
d*e - c*d*e)))/sqrt(d*e) + (45045*b^2*c^6*d^9*e^6 - (285285*b^2*c^5*d^9*e^ 
6 - (759759*b^2*c^4*d^9*e^6 + ((d*x + c)*((d*x + c)*((69069*b^2*c^6*d^9*e^ 
6 - 2860*a*b*c^4*d^11*e^6 - 240*a^2*c^2*d^13*e^6)*(d*x + c)/c^6 - 13*(2933 
7*b^2*c^7*d^9*e^6 - 1430*a*b*c^5*d^11*e^6 - 120*a^2*c^3*d^13*e^6)/c^6) + 1 
43*(6153*b^2*c^8*d^9*e^6 - 200*a*b*c^6*d^11*e^6 - 30*a^2*c^4*d^13*e^6)/c^6 
) - 429*(2534*b^2*c^9*d^9*e^6 - 30*a*b*c^7*d^11*e^6 - 15*a^2*c^5*d^13*e^6) 
/c^6)*(d*x + c))*(d*x + c))*(d*x + c))*sqrt(d*x + c)/((d*x + c)*d*e - c*d* 
e)^(13/2))*d/(e^7*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{5/2}}{{\left (e\,x\right )}^{15/2}} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x)^(5/2))/(e*x)^(15/2),x)
 

Output:

int(((a + b*x^2)^2*(c + d*x)^(5/2))/(e*x)^(15/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\frac {2 \sqrt {e}\, \left (-3465 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{6}-8505 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5} d x -5565 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d^{2} x^{2}-75 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{3} x^{3}+90 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{4} x^{4}-120 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{5} x^{5}+240 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{6} x^{6}-10010 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{6} x^{2}-27170 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} d \,x^{3}-21450 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{2} x^{4}-1430 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{3} x^{5}+2860 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{4} x^{6}-9009 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} x^{4}-33033 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d \,x^{5}-69069 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{2} x^{6}+45045 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} d^{2} x^{7}-240 \sqrt {d}\, a^{2} d^{6} x^{7}-2860 \sqrt {d}\, a b \,c^{2} d^{4} x^{7}+48279 \sqrt {d}\, b^{2} c^{4} d^{2} x^{7}\right )}{45045 c^{4} e^{8} x^{7}} \] Input:

int((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(15/2),x)
 

Output:

(2*sqrt(e)*( - 3465*sqrt(x)*sqrt(c + d*x)*a**2*c**6 - 8505*sqrt(x)*sqrt(c 
+ d*x)*a**2*c**5*d*x - 5565*sqrt(x)*sqrt(c + d*x)*a**2*c**4*d**2*x**2 - 75 
*sqrt(x)*sqrt(c + d*x)*a**2*c**3*d**3*x**3 + 90*sqrt(x)*sqrt(c + d*x)*a**2 
*c**2*d**4*x**4 - 120*sqrt(x)*sqrt(c + d*x)*a**2*c*d**5*x**5 + 240*sqrt(x) 
*sqrt(c + d*x)*a**2*d**6*x**6 - 10010*sqrt(x)*sqrt(c + d*x)*a*b*c**6*x**2 
- 27170*sqrt(x)*sqrt(c + d*x)*a*b*c**5*d*x**3 - 21450*sqrt(x)*sqrt(c + d*x 
)*a*b*c**4*d**2*x**4 - 1430*sqrt(x)*sqrt(c + d*x)*a*b*c**3*d**3*x**5 + 286 
0*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**4*x**6 - 9009*sqrt(x)*sqrt(c + d*x)*b* 
*2*c**6*x**4 - 33033*sqrt(x)*sqrt(c + d*x)*b**2*c**5*d*x**5 - 69069*sqrt(x 
)*sqrt(c + d*x)*b**2*c**4*d**2*x**6 + 45045*sqrt(d)*log((sqrt(c + d*x) + s 
qrt(x)*sqrt(d))/sqrt(c))*b**2*c**4*d**2*x**7 - 240*sqrt(d)*a**2*d**6*x**7 
- 2860*sqrt(d)*a*b*c**2*d**4*x**7 + 48279*sqrt(d)*b**2*c**4*d**2*x**7))/(4 
5045*c**4*e**8*x**7)