\(\int \frac {(c+d x)^{3/2} (a+b x^2)^2}{(e x)^{7/2}} \, dx\) [804]

Optimal result

Integrand size = 26, antiderivative size = 207 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=-\frac {b \left (b c^2-48 a d^2\right ) \sqrt {e x} \sqrt {c+d x}}{8 d e^4}-\frac {4 a b (c+d x)^{3/2}}{e^3 \sqrt {e x}}-\frac {b^2 c \sqrt {e x} (c+d x)^{3/2}}{12 d e^4}-\frac {2 a^2 (c+d x)^{5/2}}{5 c e (e x)^{5/2}}+\frac {b^2 \sqrt {e x} (c+d x)^{5/2}}{3 d e^4}-\frac {b c \left (b c^2-48 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{8 d^{3/2} e^{7/2}} \] Output:

-1/8*b*(-48*a*d^2+b*c^2)*(e*x)^(1/2)*(d*x+c)^(1/2)/d/e^4-4*a*b*(d*x+c)^(3/ 
2)/e^3/(e*x)^(1/2)-1/12*b^2*c*(e*x)^(1/2)*(d*x+c)^(3/2)/d/e^4-2/5*a^2*(d*x 
+c)^(5/2)/c/e/(e*x)^(5/2)+1/3*b^2*(e*x)^(1/2)*(d*x+c)^(5/2)/d/e^4-1/8*b*c* 
(-48*a*d^2+b*c^2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(3/ 
2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.68 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=\frac {x \left (\sqrt {d} \sqrt {c+d x} \left (240 a b c d x^2 (-2 c+d x)-48 a^2 d (c+d x)^2+5 b^2 c x^3 \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )+15 b c^2 \left (b c^2-48 a d^2\right ) x^{5/2} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )\right )}{120 c d^{3/2} (e x)^{7/2}} \] Input:

Integrate[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(7/2),x]
 

Output:

(x*(Sqrt[d]*Sqrt[c + d*x]*(240*a*b*c*d*x^2*(-2*c + d*x) - 48*a^2*d*(c + d* 
x)^2 + 5*b^2*c*x^3*(3*c^2 + 14*c*d*x + 8*d^2*x^2)) + 15*b*c^2*(b*c^2 - 48* 
a*d^2)*x^(5/2)*Log[-(Sqrt[d]*Sqrt[x]) + Sqrt[c + d*x]]))/(120*c*d^(3/2)*(e 
*x)^(7/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {520, 9, 27, 520, 27, 90, 60, 60, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^{3/2}}{(e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 9

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {b \left (\frac {\frac {b c \sqrt {e x} (c+d x)^{5/2}}{3 d e}-\frac {\left (b c^2-48 a d^2\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {e x}}dx}{6 d}}{c e}-\frac {4 a (c+d x)^{5/2}}{c e \sqrt {e x}}\right )}{e^2}-\frac {2 a^2 (c+d x)^{5/2}}{5 c e (e x)^{5/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {b \left (\frac {\frac {b c \sqrt {e x} (c+d x)^{5/2}}{3 d e}-\frac {\left (b c^2-48 a d^2\right ) \left (\frac {3}{4} c \int \frac {\sqrt {c+d x}}{\sqrt {e x}}dx+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{6 d}}{c e}-\frac {4 a (c+d x)^{5/2}}{c e \sqrt {e x}}\right )}{e^2}-\frac {2 a^2 (c+d x)^{5/2}}{5 c e (e x)^{5/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {b \left (\frac {\frac {b c \sqrt {e x} (c+d x)^{5/2}}{3 d e}-\frac {\left (b c^2-48 a d^2\right ) \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{6 d}}{c e}-\frac {4 a (c+d x)^{5/2}}{c e \sqrt {e x}}\right )}{e^2}-\frac {2 a^2 (c+d x)^{5/2}}{5 c e (e x)^{5/2}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {b \left (\frac {\frac {b c \sqrt {e x} (c+d x)^{5/2}}{3 d e}-\frac {\left (b c^2-48 a d^2\right ) \left (\frac {3}{4} c \left (c \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{6 d}}{c e}-\frac {4 a (c+d x)^{5/2}}{c e \sqrt {e x}}\right )}{e^2}-\frac {2 a^2 (c+d x)^{5/2}}{5 c e (e x)^{5/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {\frac {b c \sqrt {e x} (c+d x)^{5/2}}{3 d e}-\frac {\left (b c^2-48 a d^2\right ) \left (\frac {3}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )}{6 d}}{c e}-\frac {4 a (c+d x)^{5/2}}{c e \sqrt {e x}}\right )}{e^2}-\frac {2 a^2 (c+d x)^{5/2}}{5 c e (e x)^{5/2}}\)

Input:

Int[((c + d*x)^(3/2)*(a + b*x^2)^2)/(e*x)^(7/2),x]
 

Output:

(-2*a^2*(c + d*x)^(5/2))/(5*c*e*(e*x)^(5/2)) + (b*((-4*a*(c + d*x)^(5/2))/ 
(c*e*Sqrt[e*x]) + ((b*c*Sqrt[e*x]*(c + d*x)^(5/2))/(3*d*e) - ((b*c^2 - 48* 
a*d^2)*((Sqrt[e*x]*(c + d*x)^(3/2))/(2*e) + (3*c*((Sqrt[e*x]*Sqrt[c + d*x] 
)/e + (c*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/(Sqrt[d]*Sq 
rt[e])))/4))/(6*d))/(c*e)))/e^2
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, 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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[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 90

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

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]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {\sqrt {d x +c}\, \left (-40 b^{2} c \,d^{2} x^{5}-70 b^{2} c^{2} d \,x^{4}-240 a b c \,d^{2} x^{3}-15 b^{2} c^{3} x^{3}+48 a^{2} d^{3} x^{2}+480 a b \,c^{2} x^{2} d +96 a^{2} c \,d^{2} x +48 a^{2} c^{2} d \right )}{120 c \,x^{2} d \,e^{3} \sqrt {e x}}+\frac {b c \left (48 a \,d^{2}-b \,c^{2}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{16 d \sqrt {d e}\, e^{3} \sqrt {e x}\, \sqrt {d x +c}}\)	\(196\)
default	\(\frac {\sqrt {d x +c}\, \left (80 b^{2} c \,d^{2} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+720 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{2} d^{2} e \,x^{3}-15 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{4} e \,x^{3}+140 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{2} d \,x^{4}+480 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a b c \,d^{2} x^{3}+30 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{3} x^{3}-96 a^{2} d^{3} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-960 a b \,c^{2} d \,x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-192 a^{2} c \,d^{2} x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-96 a^{2} c^{2} d \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\right )}{240 e^{3} x^{2} c d \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(344\)

Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/120*(d*x+c)^(1/2)*(-40*b^2*c*d^2*x^5-70*b^2*c^2*d*x^4-240*a*b*c*d^2*x^3 
-15*b^2*c^3*x^3+48*a^2*d^3*x^2+480*a*b*c^2*d*x^2+96*a^2*c*d^2*x+48*a^2*c^2 
*d)/c/x^2/d/e^3/(e*x)^(1/2)+1/16*b*c*(48*a*d^2-b*c^2)/d*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^3*((d*x+c)*e*x)^(1/2)/(e 
*x)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.74 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=\left [-\frac {15 \, {\left (b^{2} c^{4} - 48 \, a b c^{2} d^{2}\right )} \sqrt {d e} x^{3} \log \left (2 \, d e x + c e + 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right ) - 2 \, {\left (40 \, b^{2} c d^{3} x^{5} + 70 \, b^{2} c^{2} d^{2} x^{4} - 96 \, a^{2} c d^{3} x - 48 \, a^{2} c^{2} d^{2} + 15 \, {\left (b^{2} c^{3} d + 16 \, a b c d^{3}\right )} x^{3} - 48 \, {\left (10 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{240 \, c d^{2} e^{4} x^{3}}, \frac {15 \, {\left (b^{2} c^{4} - 48 \, a b c^{2} d^{2}\right )} \sqrt {-d e} x^{3} \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right ) + {\left (40 \, b^{2} c d^{3} x^{5} + 70 \, b^{2} c^{2} d^{2} x^{4} - 96 \, a^{2} c d^{3} x - 48 \, a^{2} c^{2} d^{2} + 15 \, {\left (b^{2} c^{3} d + 16 \, a b c d^{3}\right )} x^{3} - 48 \, {\left (10 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{120 \, c d^{2} e^{4} x^{3}}\right ] \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(7/2),x, algorithm="fricas")
 

Output:

[-1/240*(15*(b^2*c^4 - 48*a*b*c^2*d^2)*sqrt(d*e)*x^3*log(2*d*e*x + c*e + 2 
*sqrt(d*e)*sqrt(d*x + c)*sqrt(e*x)) - 2*(40*b^2*c*d^3*x^5 + 70*b^2*c^2*d^2 
*x^4 - 96*a^2*c*d^3*x - 48*a^2*c^2*d^2 + 15*(b^2*c^3*d + 16*a*b*c*d^3)*x^3 
 - 48*(10*a*b*c^2*d^2 + a^2*d^4)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(c*d^2*e^4* 
x^3), 1/120*(15*(b^2*c^4 - 48*a*b*c^2*d^2)*sqrt(-d*e)*x^3*arctan(sqrt(-d*e 
)*sqrt(d*x + c)*sqrt(e*x)/(d*e*x + c*e)) + (40*b^2*c*d^3*x^5 + 70*b^2*c^2* 
d^2*x^4 - 96*a^2*c*d^3*x - 48*a^2*c^2*d^2 + 15*(b^2*c^3*d + 16*a*b*c*d^3)* 
x^3 - 48*(10*a*b*c^2*d^2 + a^2*d^4)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(c*d^2*e 
^4*x^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (190) = 380\).

Time = 33.16 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=- \frac {2 a^{2} c \sqrt {d} \sqrt {\frac {c}{d x} + 1}}{5 e^{\frac {7}{2}} x^{2}} - \frac {4 a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}}{5 e^{\frac {7}{2}} x} - \frac {2 a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x} + 1}}{5 c e^{\frac {7}{2}}} - \frac {4 a b c^{\frac {3}{2}}}{e^{\frac {7}{2}} \sqrt {x} \sqrt {1 + \frac {d x}{c}}} + \frac {2 a b \sqrt {c} d \sqrt {x} \sqrt {1 + \frac {d x}{c}}}{e^{\frac {7}{2}}} - \frac {4 a b \sqrt {c} d \sqrt {x}}{e^{\frac {7}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {6 a b c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{e^{\frac {7}{2}}} + \frac {b^{2} c^{\frac {5}{2}} \sqrt {x}}{8 d e^{\frac {7}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {17 b^{2} c^{\frac {3}{2}} x^{\frac {3}{2}}}{24 e^{\frac {7}{2}} \sqrt {1 + \frac {d x}{c}}} + \frac {11 b^{2} \sqrt {c} d x^{\frac {5}{2}}}{12 e^{\frac {7}{2}} \sqrt {1 + \frac {d x}{c}}} - \frac {b^{2} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{8 d^{\frac {3}{2}} e^{\frac {7}{2}}} + \frac {b^{2} d^{2} x^{\frac {7}{2}}}{3 \sqrt {c} e^{\frac {7}{2}} \sqrt {1 + \frac {d x}{c}}} \] Input:

integrate((d*x+c)**(3/2)*(b*x**2+a)**2/(e*x)**(7/2),x)
 

Output:

-2*a**2*c*sqrt(d)*sqrt(c/(d*x) + 1)/(5*e**(7/2)*x**2) - 4*a**2*d**(3/2)*sq 
rt(c/(d*x) + 1)/(5*e**(7/2)*x) - 2*a**2*d**(5/2)*sqrt(c/(d*x) + 1)/(5*c*e* 
*(7/2)) - 4*a*b*c**(3/2)/(e**(7/2)*sqrt(x)*sqrt(1 + d*x/c)) + 2*a*b*sqrt(c 
)*d*sqrt(x)*sqrt(1 + d*x/c)/e**(7/2) - 4*a*b*sqrt(c)*d*sqrt(x)/(e**(7/2)*s 
qrt(1 + d*x/c)) + 6*a*b*c*sqrt(d)*asinh(sqrt(d)*sqrt(x)/sqrt(c))/e**(7/2) 
+ b**2*c**(5/2)*sqrt(x)/(8*d*e**(7/2)*sqrt(1 + d*x/c)) + 17*b**2*c**(3/2)* 
x**(3/2)/(24*e**(7/2)*sqrt(1 + d*x/c)) + 11*b**2*sqrt(c)*d*x**(5/2)/(12*e* 
*(7/2)*sqrt(1 + d*x/c)) - b**2*c**3*asinh(sqrt(d)*sqrt(x)/sqrt(c))/(8*d**( 
3/2)*e**(7/2)) + b**2*d**2*x**(7/2)/(3*sqrt(c)*e**(7/2)*sqrt(1 + d*x/c))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(7/2),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 = 0.22 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.52 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=\frac {d {\left (\frac {15 \, {\left (b^{2} c^{3} - 48 \, a b c d^{2}\right )} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d} + \frac {{\left ({\left ({\left (5 \, {\left (2 \, {\left (4 \, {\left (d x + c\right )} b^{2} d e^{2} - 13 \, b^{2} c d e^{2}\right )} {\left (d x + c\right )} + \frac {3 \, {\left (9 \, b^{2} c^{4} d^{3} e^{4} + 16 \, a b c^{2} d^{5} e^{4}\right )}}{c^{2} d^{2} e^{2}}\right )} {\left (d x + c\right )} - \frac {25 \, b^{2} c^{5} d^{3} e^{4} + 1200 \, a b c^{3} d^{5} e^{4} + 48 \, a^{2} c d^{7} e^{4}}{c^{2} d^{2} e^{2}}\right )} {\left (d x + c\right )} - \frac {35 \, {\left (b^{2} c^{6} d^{3} e^{4} - 48 \, a b c^{4} d^{5} e^{4}\right )}}{c^{2} d^{2} e^{2}}\right )} {\left (d x + c\right )} + \frac {15 \, {\left (b^{2} c^{7} d^{3} e^{4} - 48 \, a b c^{5} d^{5} e^{4}\right )}}{c^{2} d^{2} e^{2}}\right )} \sqrt {d x + c}}{{\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {5}{2}}}\right )}}{120 \, e^{3} {\left | d \right |}} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(7/2),x, algorithm="giac")
 

Output:

1/120*d*(15*(b^2*c^3 - 48*a*b*c*d^2)*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sq 
rt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d) + (((5*(2*(4*(d*x + c)*b^2*d*e^2 
 - 13*b^2*c*d*e^2)*(d*x + c) + 3*(9*b^2*c^4*d^3*e^4 + 16*a*b*c^2*d^5*e^4)/ 
(c^2*d^2*e^2))*(d*x + c) - (25*b^2*c^5*d^3*e^4 + 1200*a*b*c^3*d^5*e^4 + 48 
*a^2*c*d^7*e^4)/(c^2*d^2*e^2))*(d*x + c) - 35*(b^2*c^6*d^3*e^4 - 48*a*b*c^ 
4*d^5*e^4)/(c^2*d^2*e^2))*(d*x + c) + 15*(b^2*c^7*d^3*e^4 - 48*a*b*c^5*d^5 
*e^4)/(c^2*d^2*e^2))*sqrt(d*x + c)/((d*x + c)*d*e - c*d*e)^(5/2))/(e^3*abs 
(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{7/2}} \,d x \] Input:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(7/2),x)
 

Output:

int(((a + b*x^2)^2*(c + d*x)^(3/2))/(e*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.35 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )^2}{(e x)^{7/2}} \, dx=\frac {\sqrt {e}\, \left (-96 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{2}-192 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{3} x -96 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{4} x^{2}-960 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{2} x^{2}+480 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{3} x^{3}+30 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d \,x^{3}+140 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{2} x^{4}+80 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{3} x^{5}+1440 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{2} d^{2} x^{3}-30 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{4} x^{3}-96 \sqrt {d}\, a^{2} d^{4} x^{3}+792 \sqrt {d}\, a b \,c^{2} d^{2} x^{3}-13 \sqrt {d}\, b^{2} c^{4} x^{3}\right )}{240 c \,d^{2} e^{4} x^{3}} \] Input:

int((d*x+c)^(3/2)*(b*x^2+a)^2/(e*x)^(7/2),x)
 

Output:

(sqrt(e)*( - 96*sqrt(x)*sqrt(c + d*x)*a**2*c**2*d**2 - 192*sqrt(x)*sqrt(c 
+ d*x)*a**2*c*d**3*x - 96*sqrt(x)*sqrt(c + d*x)*a**2*d**4*x**2 - 960*sqrt( 
x)*sqrt(c + d*x)*a*b*c**2*d**2*x**2 + 480*sqrt(x)*sqrt(c + d*x)*a*b*c*d**3 
*x**3 + 30*sqrt(x)*sqrt(c + d*x)*b**2*c**3*d*x**3 + 140*sqrt(x)*sqrt(c + d 
*x)*b**2*c**2*d**2*x**4 + 80*sqrt(x)*sqrt(c + d*x)*b**2*c*d**3*x**5 + 1440 
*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c**2*d**2*x**3 
 - 30*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2*c**4*x** 
3 - 96*sqrt(d)*a**2*d**4*x**3 + 792*sqrt(d)*a*b*c**2*d**2*x**3 - 13*sqrt(d 
)*b**2*c**4*x**3))/(240*c*d**2*e**4*x**3)