\(\int \frac {(c+d x)^{3/2} (a+b x^2)}{(e x)^{9/2}} \, dx\) [758]

Optimal result

Integrand size = 24, antiderivative size = 147 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=-\frac {2 b c \sqrt {c+d x}}{3 e^3 (e x)^{3/2}}-\frac {8 b d \sqrt {c+d x}}{3 e^4 \sqrt {e x}}-\frac {2 a (c+d x)^{5/2}}{7 c e (e x)^{7/2}}+\frac {4 a d (c+d x)^{5/2}}{35 c^2 e^2 (e x)^{5/2}}+\frac {2 b d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{e^{9/2}} \] Output:

-2/3*b*c*(d*x+c)^(1/2)/e^3/(e*x)^(3/2)-8/3*b*d*(d*x+c)^(1/2)/e^4/(e*x)^(1/ 
2)-2/7*a*(d*x+c)^(5/2)/c/e/(e*x)^(7/2)+4/35*a*d*(d*x+c)^(5/2)/c^2/e^2/(e*x 
)^(5/2)+2*b*d^(3/2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/e^( 
9/2)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.71 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=-\frac {2 \sqrt {e x} \left (\sqrt {c+d x} \left (3 a (5 c-2 d x) (c+d x)^2+35 b c^2 x^2 (c+4 d x)\right )+105 b c^2 d^{3/2} x^{7/2} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )\right )}{105 c^2 e^5 x^4} \] Input:

Integrate[((c + d*x)^(3/2)*(a + b*x^2))/(e*x)^(9/2),x]
 

Output:

(-2*Sqrt[e*x]*(Sqrt[c + d*x]*(3*a*(5*c - 2*d*x)*(c + d*x)^2 + 35*b*c^2*x^2 
*(c + 4*d*x)) + 105*b*c^2*d^(3/2)*x^(7/2)*Log[-(Sqrt[d]*Sqrt[x]) + Sqrt[c 
+ d*x]]))/(105*c^2*e^5*x^4)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {520, 27, 87, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right ) (c+d x)^{3/2}}{(e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {7 b c \int \frac {(c+d x)^{3/2}}{(e x)^{5/2}}dx}{e}-\frac {4 a d (c+d x)^{5/2}}{5 c e (e x)^{5/2}}}{7 c e}-\frac {2 a (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\)

\(\Big \downarrow \) 57

\(\displaystyle -\frac {-\frac {7 b c \left (\frac {d \int \frac {\sqrt {c+d x}}{(e x)^{3/2}}dx}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {4 a d (c+d x)^{5/2}}{5 c e (e x)^{5/2}}}{7 c e}-\frac {2 a (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\)

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 65

\(\displaystyle -\frac {-\frac {7 b c \left (\frac {d \left (\frac {2 d \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{e}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}-\frac {4 a d (c+d x)^{5/2}}{5 c e (e x)^{5/2}}}{7 c e}-\frac {2 a (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {4 a d (c+d x)^{5/2}}{5 c e (e x)^{5/2}}-\frac {7 b c \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{e^{3/2}}-\frac {2 \sqrt {c+d x}}{e \sqrt {e x}}\right )}{e}-\frac {2 (c+d x)^{3/2}}{3 e (e x)^{3/2}}\right )}{e}}{7 c e}-\frac {2 a (c+d x)^{5/2}}{7 c e (e x)^{7/2}}\)

Input:

Int[((c + d*x)^(3/2)*(a + b*x^2))/(e*x)^(9/2),x]
 

Output:

(-2*a*(c + d*x)^(5/2))/(7*c*e*(e*x)^(7/2)) - ((-4*a*d*(c + d*x)^(5/2))/(5* 
c*e*(e*x)^(5/2)) - (7*b*c*((-2*(c + d*x)^(3/2))/(3*e*(e*x)^(3/2)) + (d*((- 
2*Sqrt[c + d*x])/(e*Sqrt[e*x]) + (2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(S 
qrt[e]*Sqrt[c + d*x])])/e^(3/2)))/e))/e)/(7*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]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-6 a \,x^{3} d^{3}+140 b \,c^{2} d \,x^{3}+3 a \,d^{2} x^{2} c +35 b \,c^{3} x^{2}+24 a d x \,c^{2}+15 c^{3} a \right )}{105 x^{3} c^{2} e^{4} \sqrt {e x}}+\frac {b \,d^{2} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{\sqrt {d e}\, e^{4} \sqrt {e x}\, \sqrt {d x +c}}\)	\(143\)
default	\(\frac {\sqrt {d x +c}\, \left (105 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b \,c^{2} d^{2} e \,x^{4}+12 a \,d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-280 b \,c^{2} d \,x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-6 a c \,d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-70 b \,c^{3} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-48 a \,c^{2} d x \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-30 a \,c^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\right )}{105 e^{4} x^{3} c^{2} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(229\)

Input:

int((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-2/105*(d*x+c)^(1/2)*(-6*a*d^3*x^3+140*b*c^2*d*x^3+3*a*c*d^2*x^2+35*b*c^3* 
x^2+24*a*c^2*d*x+15*a*c^3)/x^3/c^2/e^4/(e*x)^(1/2)+b*d^2*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^4*((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 = 243, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=\left [\frac {105 \, b c^{2} d e x^{4} \sqrt {\frac {d}{e}} \log \left (2 \, d x + 2 \, \sqrt {d x + c} \sqrt {e x} \sqrt {\frac {d}{e}} + c\right ) - 2 \, {\left (24 \, a c^{2} d x + 15 \, a c^{3} + 2 \, {\left (70 \, b c^{2} d - 3 \, a d^{3}\right )} x^{3} + {\left (35 \, b c^{3} + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{105 \, c^{2} e^{5} x^{4}}, -\frac {2 \, {\left (105 \, b c^{2} d e x^{4} \sqrt {-\frac {d}{e}} \arctan \left (\frac {\sqrt {e x} \sqrt {-\frac {d}{e}}}{\sqrt {d x + c}}\right ) + {\left (24 \, a c^{2} d x + 15 \, a c^{3} + 2 \, {\left (70 \, b c^{2} d - 3 \, a d^{3}\right )} x^{3} + {\left (35 \, b c^{3} + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}\right )}}{105 \, c^{2} e^{5} x^{4}}\right ] \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(9/2),x, algorithm="fricas")
 

Output:

[1/105*(105*b*c^2*d*e*x^4*sqrt(d/e)*log(2*d*x + 2*sqrt(d*x + c)*sqrt(e*x)* 
sqrt(d/e) + c) - 2*(24*a*c^2*d*x + 15*a*c^3 + 2*(70*b*c^2*d - 3*a*d^3)*x^3 
 + (35*b*c^3 + 3*a*c*d^2)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(c^2*e^5*x^4), -2/ 
105*(105*b*c^2*d*e*x^4*sqrt(-d/e)*arctan(sqrt(e*x)*sqrt(-d/e)/sqrt(d*x + c 
)) + (24*a*c^2*d*x + 15*a*c^3 + 2*(70*b*c^2*d - 3*a*d^3)*x^3 + (35*b*c^3 + 
 3*a*c*d^2)*x^2)*sqrt(d*x + c)*sqrt(e*x))/(c^2*e^5*x^4)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (143) = 286\).

Time = 67.01 (sec) , antiderivative size = 685, normalized size of antiderivative = 4.66 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=- \frac {30 a c^{6} d^{\frac {9}{2}} \sqrt {\frac {c}{d x} + 1}}{105 c^{5} d^{4} e^{\frac {9}{2}} x^{3} + 210 c^{4} d^{5} e^{\frac {9}{2}} x^{4} + 105 c^{3} d^{6} e^{\frac {9}{2}} x^{5}} - \frac {66 a c^{5} d^{\frac {11}{2}} x \sqrt {\frac {c}{d x} + 1}}{105 c^{5} d^{4} e^{\frac {9}{2}} x^{3} + 210 c^{4} d^{5} e^{\frac {9}{2}} x^{4} + 105 c^{3} d^{6} e^{\frac {9}{2}} x^{5}} - \frac {34 a c^{4} d^{\frac {13}{2}} x^{2} \sqrt {\frac {c}{d x} + 1}}{105 c^{5} d^{4} e^{\frac {9}{2}} x^{3} + 210 c^{4} d^{5} e^{\frac {9}{2}} x^{4} + 105 c^{3} d^{6} e^{\frac {9}{2}} x^{5}} - \frac {6 a c^{3} d^{\frac {15}{2}} x^{3} \sqrt {\frac {c}{d x} + 1}}{105 c^{5} d^{4} e^{\frac {9}{2}} x^{3} + 210 c^{4} d^{5} e^{\frac {9}{2}} x^{4} + 105 c^{3} d^{6} e^{\frac {9}{2}} x^{5}} - \frac {24 a c^{2} d^{\frac {17}{2}} x^{4} \sqrt {\frac {c}{d x} + 1}}{105 c^{5} d^{4} e^{\frac {9}{2}} x^{3} + 210 c^{4} d^{5} e^{\frac {9}{2}} x^{4} + 105 c^{3} d^{6} e^{\frac {9}{2}} x^{5}} - \frac {16 a c d^{\frac {19}{2}} x^{5} \sqrt {\frac {c}{d x} + 1}}{105 c^{5} d^{4} e^{\frac {9}{2}} x^{3} + 210 c^{4} d^{5} e^{\frac {9}{2}} x^{4} + 105 c^{3} d^{6} e^{\frac {9}{2}} x^{5}} - \frac {2 a d^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}}{5 e^{\frac {9}{2}} x^{2}} - \frac {2 a d^{\frac {5}{2}} \sqrt {\frac {c}{d x} + 1}}{15 c e^{\frac {9}{2}} x} + \frac {4 a d^{\frac {7}{2}} \sqrt {\frac {c}{d x} + 1}}{15 c^{2} e^{\frac {9}{2}}} - \frac {2 b \sqrt {c} d}{e^{\frac {9}{2}} \sqrt {x} \sqrt {1 + \frac {d x}{c}}} - \frac {2 b c \sqrt {d} \sqrt {\frac {c}{d x} + 1}}{3 e^{\frac {9}{2}} x} - \frac {2 b d^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}}{3 e^{\frac {9}{2}}} + \frac {2 b d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}} \right )}}{e^{\frac {9}{2}}} - \frac {2 b d^{2} \sqrt {x}}{\sqrt {c} e^{\frac {9}{2}} \sqrt {1 + \frac {d x}{c}}} \] Input:

integrate((d*x+c)**(3/2)*(b*x**2+a)/(e*x)**(9/2),x)
 

Output:

-30*a*c**6*d**(9/2)*sqrt(c/(d*x) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 210*c 
**4*d**5*e**(9/2)*x**4 + 105*c**3*d**6*e**(9/2)*x**5) - 66*a*c**5*d**(11/2 
)*x*sqrt(c/(d*x) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 210*c**4*d**5*e**(9/2 
)*x**4 + 105*c**3*d**6*e**(9/2)*x**5) - 34*a*c**4*d**(13/2)*x**2*sqrt(c/(d 
*x) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 210*c**4*d**5*e**(9/2)*x**4 + 105* 
c**3*d**6*e**(9/2)*x**5) - 6*a*c**3*d**(15/2)*x**3*sqrt(c/(d*x) + 1)/(105* 
c**5*d**4*e**(9/2)*x**3 + 210*c**4*d**5*e**(9/2)*x**4 + 105*c**3*d**6*e**( 
9/2)*x**5) - 24*a*c**2*d**(17/2)*x**4*sqrt(c/(d*x) + 1)/(105*c**5*d**4*e** 
(9/2)*x**3 + 210*c**4*d**5*e**(9/2)*x**4 + 105*c**3*d**6*e**(9/2)*x**5) - 
16*a*c*d**(19/2)*x**5*sqrt(c/(d*x) + 1)/(105*c**5*d**4*e**(9/2)*x**3 + 210 
*c**4*d**5*e**(9/2)*x**4 + 105*c**3*d**6*e**(9/2)*x**5) - 2*a*d**(3/2)*sqr 
t(c/(d*x) + 1)/(5*e**(9/2)*x**2) - 2*a*d**(5/2)*sqrt(c/(d*x) + 1)/(15*c*e* 
*(9/2)*x) + 4*a*d**(7/2)*sqrt(c/(d*x) + 1)/(15*c**2*e**(9/2)) - 2*b*sqrt(c 
)*d/(e**(9/2)*sqrt(x)*sqrt(1 + d*x/c)) - 2*b*c*sqrt(d)*sqrt(c/(d*x) + 1)/( 
3*e**(9/2)*x) - 2*b*d**(3/2)*sqrt(c/(d*x) + 1)/(3*e**(9/2)) + 2*b*d**(3/2) 
*asinh(sqrt(d)*sqrt(x)/sqrt(c))/e**(9/2) - 2*b*d**2*sqrt(x)/(sqrt(c)*e**(9 
/2)*sqrt(1 + d*x/c))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(9/2),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 = 0.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.27 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=-\frac {2 \, d^{5} {\left (\frac {105 \, b \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{2}} - \frac {{\left (105 \, b c^{3} d e^{3} - {\left (350 \, b c^{2} d e^{3} + {\left (d x + c\right )} {\left (\frac {2 \, {\left (70 \, b c^{3} d^{3} e^{3} - 3 \, a c d^{5} e^{3}\right )} {\left (d x + c\right )}}{c^{3} d^{2}} - \frac {7 \, {\left (55 \, b c^{4} d^{3} e^{3} - 3 \, a c^{2} d^{5} e^{3}\right )}}{c^{3} d^{2}}\right )}\right )} {\left (d x + c\right )}\right )} \sqrt {d x + c}}{{\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {7}{2}}}\right )}}{105 \, e^{4} {\left | d \right |}} \] Input:

integrate((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(9/2),x, algorithm="giac")
 

Output:

-2/105*d^5*(105*b*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - 
c*d*e)))/(sqrt(d*e)*d^2) - (105*b*c^3*d*e^3 - (350*b*c^2*d*e^3 + (d*x + c) 
*(2*(70*b*c^3*d^3*e^3 - 3*a*c*d^5*e^3)*(d*x + c)/(c^3*d^2) - 7*(55*b*c^4*d 
^3*e^3 - 3*a*c^2*d^5*e^3)/(c^3*d^2)))*(d*x + c))*sqrt(d*x + c)/((d*x + c)* 
d*e - c*d*e)^(7/2))/(e^4*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,{\left (c+d\,x\right )}^{3/2}}{{\left (e\,x\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)*(c + d*x)^(3/2))/(e*x)^(9/2),x)
 

Output:

int(((a + b*x^2)*(c + d*x)^(3/2))/(e*x)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14 \[ \int \frac {(c+d x)^{3/2} \left (a+b x^2\right )}{(e x)^{9/2}} \, dx=\frac {2 \sqrt {e}\, \left (-15 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{3}-24 \sqrt {x}\, \sqrt {d x +c}\, a \,c^{2} d x -3 \sqrt {x}\, \sqrt {d x +c}\, a c \,d^{2} x^{2}+6 \sqrt {x}\, \sqrt {d x +c}\, a \,d^{3} x^{3}-35 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{3} x^{2}-140 \sqrt {x}\, \sqrt {d x +c}\, b \,c^{2} d \,x^{3}+105 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b \,c^{2} d \,x^{4}-6 \sqrt {d}\, a \,d^{3} x^{4}+80 \sqrt {d}\, b \,c^{2} d \,x^{4}\right )}{105 c^{2} e^{5} x^{4}} \] Input:

int((d*x+c)^(3/2)*(b*x^2+a)/(e*x)^(9/2),x)
 

Output:

(2*sqrt(e)*( - 15*sqrt(x)*sqrt(c + d*x)*a*c**3 - 24*sqrt(x)*sqrt(c + d*x)* 
a*c**2*d*x - 3*sqrt(x)*sqrt(c + d*x)*a*c*d**2*x**2 + 6*sqrt(x)*sqrt(c + d* 
x)*a*d**3*x**3 - 35*sqrt(x)*sqrt(c + d*x)*b*c**3*x**2 - 140*sqrt(x)*sqrt(c 
 + d*x)*b*c**2*d*x**3 + 105*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/ 
sqrt(c))*b*c**2*d*x**4 - 6*sqrt(d)*a*d**3*x**4 + 80*sqrt(d)*b*c**2*d*x**4) 
)/(105*c**2*e**5*x**4)