\(\int \frac {(e x)^{7/2} (a+b x^2)}{(c+d x)^{5/2}} \, dx\) [780]

Optimal result

Integrand size = 24, antiderivative size = 264 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=-\frac {\left (11 b c^2+8 a d^2\right ) (e x)^{7/2}}{12 d^3 (c+d x)^{3/2}}+\frac {b (e x)^{11/2}}{4 d e^2 (c+d x)^{3/2}}-\frac {\left (55 b c^2+28 a d^2\right ) e (e x)^{5/2}}{6 d^4 \sqrt {c+d x}}-\frac {35 c \left (33 b c^2+16 a d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x}}{64 d^6}+\frac {35 \left (33 b c^2+16 a d^2\right ) e^2 (e x)^{3/2} \sqrt {c+d x}}{96 d^5}-\frac {11 b c e (e x)^{5/2} \sqrt {c+d x}}{24 d^4}+\frac {35 c^2 \left (33 b c^2+16 a d^2\right ) e^{7/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{64 d^{13/2}} \] Output:

-1/12*(8*a*d^2+11*b*c^2)*(e*x)^(7/2)/d^3/(d*x+c)^(3/2)+1/4*b*(e*x)^(11/2)/ 
d/e^2/(d*x+c)^(3/2)-1/6*(28*a*d^2+55*b*c^2)*e*(e*x)^(5/2)/d^4/(d*x+c)^(1/2 
)-35/64*c*(16*a*d^2+33*b*c^2)*e^3*(e*x)^(1/2)*(d*x+c)^(1/2)/d^6+35/96*(16* 
a*d^2+33*b*c^2)*e^2*(e*x)^(3/2)*(d*x+c)^(1/2)/d^5-11/24*b*c*e*(e*x)^(5/2)* 
(d*x+c)^(1/2)/d^4+35/64*c^2*(16*a*d^2+33*b*c^2)*e^(7/2)*arctanh(d^(1/2)*(e 
*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(13/2)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.69 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {e^3 \sqrt {e x} \left (\frac {\sqrt {d} \left (-16 a d^2 \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )-b \left (3465 c^5+4620 c^4 d x+693 c^3 d^2 x^2-198 c^2 d^3 x^3+88 c d^4 x^4-48 d^5 x^5\right )\right )}{(c+d x)^{3/2}}+\frac {210 c^2 \left (33 b c^2+16 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )}{\sqrt {x}}\right )}{192 d^{13/2}} \] Input:

Integrate[((e*x)^(7/2)*(a + b*x^2))/(c + d*x)^(5/2),x]
 

Output:

(e^3*Sqrt[e*x]*((Sqrt[d]*(-16*a*d^2*(105*c^3 + 140*c^2*d*x + 21*c*d^2*x^2 
- 6*d^3*x^3) - b*(3465*c^5 + 4620*c^4*d*x + 693*c^3*d^2*x^2 - 198*c^2*d^3* 
x^3 + 88*c*d^4*x^4 - 48*d^5*x^5)))/(c + d*x)^(3/2) + (210*c^2*(33*b*c^2 + 
16*a*d^2)*ArcTanh[(Sqrt[d]*Sqrt[x])/(-Sqrt[c] + Sqrt[c + d*x])])/Sqrt[x])) 
/(192*d^(13/2))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {519, 27, 87, 60, 60, 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.

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

\(\Big \downarrow \) 519

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {2 \int \frac {3 (e x)^{7/2} \left (\left (\frac {3 b c^2}{d^2}+2 a\right ) d-b c x\right )}{2 d (c+d x)^{3/2}}dx}{3 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\int \frac {(e x)^{7/2} \left (\frac {3 b c^2}{d}-b x c+2 a d\right )}{(c+d x)^{3/2}}dx}{c d}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\frac {4 (e x)^{9/2} \left (a d^2+2 b c^2\right )}{c d e \sqrt {c+d x}}-\left (\frac {16 a d}{c}+\frac {33 b c}{d}\right ) \int \frac {(e x)^{7/2}}{\sqrt {c+d x}}dx}{c d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\frac {4 (e x)^{9/2} \left (a d^2+2 b c^2\right )}{c d e \sqrt {c+d x}}-\left (\frac {16 a d}{c}+\frac {33 b c}{d}\right ) \left (\frac {(e x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 c e \int \frac {(e x)^{5/2}}{\sqrt {c+d x}}dx}{8 d}\right )}{c d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\frac {4 (e x)^{9/2} \left (a d^2+2 b c^2\right )}{c d e \sqrt {c+d x}}-\left (\frac {16 a d}{c}+\frac {33 b c}{d}\right ) \left (\frac {(e x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 c e \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 d}\right )}{c d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\frac {4 (e x)^{9/2} \left (a d^2+2 b c^2\right )}{c d e \sqrt {c+d x}}-\left (\frac {16 a d}{c}+\frac {33 b c}{d}\right ) \left (\frac {(e x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 c e \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 d}\right )}{c d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\frac {4 (e x)^{9/2} \left (a d^2+2 b c^2\right )}{c d e \sqrt {c+d x}}-\left (\frac {16 a d}{c}+\frac {33 b c}{d}\right ) \left (\frac {(e x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 c e \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 d}\right )}{c d}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\frac {4 (e x)^{9/2} \left (a d^2+2 b c^2\right )}{c d e \sqrt {c+d x}}-\left (\frac {16 a d}{c}+\frac {33 b c}{d}\right ) \left (\frac {(e x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 c e \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 d}\right )}{c d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 (e x)^{9/2} \left (a+\frac {b c^2}{d^2}\right )}{3 c e (c+d x)^{3/2}}-\frac {\frac {4 (e x)^{9/2} \left (a d^2+2 b c^2\right )}{c d e \sqrt {c+d x}}-\left (\frac {16 a d}{c}+\frac {33 b c}{d}\right ) \left (\frac {(e x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 c e \left (\frac {(e x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 c e \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 d}\right )}{c d}\)

Input:

Int[((e*x)^(7/2)*(a + b*x^2))/(c + d*x)^(5/2),x]
 

Output:

(2*(a + (b*c^2)/d^2)*(e*x)^(9/2))/(3*c*e*(c + d*x)^(3/2)) - ((4*(2*b*c^2 + 
 a*d^2)*(e*x)^(9/2))/(c*d*e*Sqrt[c + d*x]) - ((33*b*c)/d + (16*a*d)/c)*((( 
e*x)^(7/2)*Sqrt[c + d*x])/(4*d) - (7*c*e*(((e*x)^(5/2)*Sqrt[c + d*x])/(3*d 
) - (5*c*e*(((e*x)^(3/2)*Sqrt[c + d*x])/(2*d) - (3*c*e*((Sqrt[e*x]*Sqrt[c 
+ d*x])/d - (c*Sqrt[e]*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d*x]) 
])/d^(3/2)))/(4*d)))/(6*d)))/(8*d)))/(c*d)
 

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 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 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 519

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

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.33

method	result	size
risch	\(-\frac {\left (-48 b \,d^{3} x^{3}+184 b c \,d^{2} x^{2}-96 a x \,d^{3}-518 b \,c^{2} d x +528 a \,d^{2} c +1545 b \,c^{3}\right ) x \sqrt {d x +c}\, e^{4}}{192 d^{6} \sqrt {e x}}+\frac {c^{2} \left (\frac {560 a \,d^{2} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}+\frac {1155 b \,c^{2} \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right )}{\sqrt {d e}}-\frac {512 \left (2 a \,d^{2}+3 b \,c^{2}\right ) \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{d e \left (x +\frac {c}{d}\right )}+\frac {128 c^{2} \left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {2 \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{3 c e \left (x +\frac {c}{d}\right )^{2}}+\frac {4 d \sqrt {d e \left (x +\frac {c}{d}\right )^{2}-c e \left (x +\frac {c}{d}\right )}}{3 e \,c^{2} \left (x +\frac {c}{d}\right )}\right )}{d^{2}}\right ) e^{4} \sqrt {\left (d x +c \right ) e x}}{128 d^{6} \sqrt {e x}\, \sqrt {d x +c}}\)	\(350\)
default	\(\frac {\left (96 b \,d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-176 b c \,d^{4} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+1680 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a \,c^{2} d^{4} e \,x^{2}+3465 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b \,c^{4} d^{2} e \,x^{2}+192 a \,d^{5} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+396 b \,c^{2} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+3360 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a \,c^{3} d^{3} e x +6930 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b \,c^{5} d e x -672 a c \,d^{4} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-1386 b \,c^{3} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+1680 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a \,c^{4} d^{2} e +3465 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b \,c^{6} e -4480 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a \,c^{2} d^{3} x -9240 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b \,c^{4} d x -3360 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a \,c^{3} d^{2}-6930 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b \,c^{5}\right ) \sqrt {e x}\, e^{3}}{384 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, d^{6} \left (d x +c \right )^{\frac {3}{2}}}\)	\(549\)

Input:

int((e*x)^(7/2)*(b*x^2+a)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/192*(-48*b*d^3*x^3+184*b*c*d^2*x^2-96*a*d^3*x-518*b*c^2*d*x+528*a*c*d^2 
+1545*b*c^3)*x*(d*x+c)^(1/2)/d^6*e^4/(e*x)^(1/2)+1/128*c^2/d^6*(560*a*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)+1155*b*c 
^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)-512*( 
2*a*d^2+3*b*c^2)/d/e/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)+128*c^2*(a* 
d^2+b*c^2)/d^2*(2/3/c/e/(x+c/d)^2*(d*e*(x+c/d)^2-c*e*(x+c/d))^(1/2)+4/3*d/ 
e/c^2/(x+c/d)*(d*e*(x+c/d)^2-c*e*(x+c/d))^(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 = 0.12 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.06 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\left [\frac {105 \, {\left ({\left (33 \, b c^{4} d^{2} + 16 \, a c^{2} d^{4}\right )} e^{3} x^{2} + 2 \, {\left (33 \, b c^{5} d + 16 \, a c^{3} d^{3}\right )} e^{3} x + {\left (33 \, b c^{6} + 16 \, a c^{4} d^{2}\right )} e^{3}\right )} \sqrt {\frac {e}{d}} \log \left (2 \, d e x + 2 \, \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{d}} + c e\right ) + 2 \, {\left (48 \, b d^{5} e^{3} x^{5} - 88 \, b c d^{4} e^{3} x^{4} + 6 \, {\left (33 \, b c^{2} d^{3} + 16 \, a d^{5}\right )} e^{3} x^{3} - 21 \, {\left (33 \, b c^{3} d^{2} + 16 \, a c d^{4}\right )} e^{3} x^{2} - 140 \, {\left (33 \, b c^{4} d + 16 \, a c^{2} d^{3}\right )} e^{3} x - 105 \, {\left (33 \, b c^{5} + 16 \, a c^{3} d^{2}\right )} e^{3}\right )} \sqrt {d x + c} \sqrt {e x}}{384 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}}, -\frac {105 \, {\left ({\left (33 \, b c^{4} d^{2} + 16 \, a c^{2} d^{4}\right )} e^{3} x^{2} + 2 \, {\left (33 \, b c^{5} d + 16 \, a c^{3} d^{3}\right )} e^{3} x + {\left (33 \, b c^{6} + 16 \, a c^{4} d^{2}\right )} e^{3}\right )} \sqrt {-\frac {e}{d}} \arctan \left (\frac {\sqrt {d x + c} \sqrt {e x} d \sqrt {-\frac {e}{d}}}{d e x + c e}\right ) - {\left (48 \, b d^{5} e^{3} x^{5} - 88 \, b c d^{4} e^{3} x^{4} + 6 \, {\left (33 \, b c^{2} d^{3} + 16 \, a d^{5}\right )} e^{3} x^{3} - 21 \, {\left (33 \, b c^{3} d^{2} + 16 \, a c d^{4}\right )} e^{3} x^{2} - 140 \, {\left (33 \, b c^{4} d + 16 \, a c^{2} d^{3}\right )} e^{3} x - 105 \, {\left (33 \, b c^{5} + 16 \, a c^{3} d^{2}\right )} e^{3}\right )} \sqrt {d x + c} \sqrt {e x}}{192 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}}\right ] \] Input:

integrate((e*x)^(7/2)*(b*x^2+a)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/384*(105*((33*b*c^4*d^2 + 16*a*c^2*d^4)*e^3*x^2 + 2*(33*b*c^5*d + 16*a* 
c^3*d^3)*e^3*x + (33*b*c^6 + 16*a*c^4*d^2)*e^3)*sqrt(e/d)*log(2*d*e*x + 2* 
sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(48*b*d^5*e^3*x^5 - 88*b*c* 
d^4*e^3*x^4 + 6*(33*b*c^2*d^3 + 16*a*d^5)*e^3*x^3 - 21*(33*b*c^3*d^2 + 16* 
a*c*d^4)*e^3*x^2 - 140*(33*b*c^4*d + 16*a*c^2*d^3)*e^3*x - 105*(33*b*c^5 + 
 16*a*c^3*d^2)*e^3)*sqrt(d*x + c)*sqrt(e*x))/(d^8*x^2 + 2*c*d^7*x + c^2*d^ 
6), -1/192*(105*((33*b*c^4*d^2 + 16*a*c^2*d^4)*e^3*x^2 + 2*(33*b*c^5*d + 1 
6*a*c^3*d^3)*e^3*x + (33*b*c^6 + 16*a*c^4*d^2)*e^3)*sqrt(-e/d)*arctan(sqrt 
(d*x + c)*sqrt(e*x)*d*sqrt(-e/d)/(d*e*x + c*e)) - (48*b*d^5*e^3*x^5 - 88*b 
*c*d^4*e^3*x^4 + 6*(33*b*c^2*d^3 + 16*a*d^5)*e^3*x^3 - 21*(33*b*c^3*d^2 + 
16*a*c*d^4)*e^3*x^2 - 140*(33*b*c^4*d + 16*a*c^2*d^3)*e^3*x - 105*(33*b*c^ 
5 + 16*a*c^3*d^2)*e^3)*sqrt(d*x + c)*sqrt(e*x))/(d^8*x^2 + 2*c*d^7*x + c^2 
*d^6)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(7/2)*(b*x**2+a)/(d*x+c)**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(7/2)*(b*x^2+a)/(d*x+c)^(5/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (214) = 428\).

Time = 0.30 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.90 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {1}{192} \, \sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )} b e^{3} {\left | d \right |}}{d^{8}} - \frac {41 \, b c e^{3} {\left | d \right |}}{d^{8}}\right )} + \frac {515 \, b c^{2} d^{31} e^{3} {\left | d \right |} + 48 \, a d^{33} e^{3} {\left | d \right |}}{d^{39}}\right )} - \frac {3 \, {\left (765 \, b c^{3} d^{31} e^{3} {\left | d \right |} + 208 \, a c d^{33} e^{3} {\left | d \right |}\right )}}{d^{39}}\right )} \sqrt {d x + c} - \frac {35 \, {\left (33 \, \sqrt {d e} b c^{4} e^{3} {\left | d \right |} + 16 \, \sqrt {d e} a c^{2} d^{2} e^{3} {\left | d \right |}\right )} \log \left ({\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}{128 \, d^{8}} - \frac {8 \, {\left (8 \, \sqrt {d e} b c^{7} d^{2} e^{6} {\left | d \right |} + 5 \, \sqrt {d e} a c^{5} d^{4} e^{6} {\left | d \right |} + 15 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} b c^{6} d e^{5} {\left | d \right |} + 9 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} a c^{4} d^{3} e^{5} {\left | d \right |} + 9 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} b c^{5} e^{4} {\left | d \right |} + 6 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} a c^{3} d^{2} e^{4} {\left | d \right |}\right )}}{3 \, {\left (c d e + {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}^{3} d^{7}} \] Input:

integrate((e*x)^(7/2)*(b*x^2+a)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

1/192*sqrt((d*x + c)*d*e - c*d*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)*b 
*e^3*abs(d)/d^8 - 41*b*c*e^3*abs(d)/d^8) + (515*b*c^2*d^31*e^3*abs(d) + 48 
*a*d^33*e^3*abs(d))/d^39) - 3*(765*b*c^3*d^31*e^3*abs(d) + 208*a*c*d^33*e^ 
3*abs(d))/d^39)*sqrt(d*x + c) - 35/128*(33*sqrt(d*e)*b*c^4*e^3*abs(d) + 16 
*sqrt(d*e)*a*c^2*d^2*e^3*abs(d))*log((sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x 
+ c)*d*e - c*d*e))^2)/d^8 - 8/3*(8*sqrt(d*e)*b*c^7*d^2*e^6*abs(d) + 5*sqrt 
(d*e)*a*c^5*d^4*e^6*abs(d) + 15*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt( 
(d*x + c)*d*e - c*d*e))^2*b*c^6*d*e^5*abs(d) + 9*sqrt(d*e)*(sqrt(d*e)*sqrt 
(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^2*a*c^4*d^3*e^5*abs(d) + 9*sqrt(d 
*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e))^4*b*c^5*e^4*ab 
s(d) + 6*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c)*d*e - c*d*e)) 
^4*a*c^3*d^2*e^4*abs(d))/((c*d*e + (sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + 
c)*d*e - c*d*e))^2)^3*d^7)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,\left (b\,x^2+a\right )}{{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(((e*x)^(7/2)*(a + b*x^2))/(c + d*x)^(5/2),x)
 

Output:

int(((e*x)^(7/2)*(a + b*x^2))/(c + d*x)^(5/2), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.31 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {e}\, e^{3} \left (3360 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a \,c^{3} d^{2}+3360 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a \,c^{2} d^{3} x +6930 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b \,c^{5}+6930 \sqrt {d}\, \sqrt {d x +c}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b \,c^{4} d x +700 \sqrt {d}\, \sqrt {d x +c}\, a \,c^{3} d^{2}+700 \sqrt {d}\, \sqrt {d x +c}\, a \,c^{2} d^{3} x +1617 \sqrt {d}\, \sqrt {d x +c}\, b \,c^{5}+1617 \sqrt {d}\, \sqrt {d x +c}\, b \,c^{4} d x -3360 \sqrt {x}\, a \,c^{3} d^{3}-4480 \sqrt {x}\, a \,c^{2} d^{4} x -672 \sqrt {x}\, a c \,d^{5} x^{2}+192 \sqrt {x}\, a \,d^{6} x^{3}-6930 \sqrt {x}\, b \,c^{5} d -9240 \sqrt {x}\, b \,c^{4} d^{2} x -1386 \sqrt {x}\, b \,c^{3} d^{3} x^{2}+396 \sqrt {x}\, b \,c^{2} d^{4} x^{3}-176 \sqrt {x}\, b c \,d^{5} x^{4}+96 \sqrt {x}\, b \,d^{6} x^{5}\right )}{384 \sqrt {d x +c}\, d^{7} \left (d x +c \right )} \] Input:

int((e*x)^(7/2)*(b*x^2+a)/(d*x+c)^(5/2),x)
 

Output:

(sqrt(e)*e**3*(3360*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d*x) + sqrt(x)*sqr 
t(d))/sqrt(c))*a*c**3*d**2 + 3360*sqrt(d)*sqrt(c + d*x)*log((sqrt(c + d*x) 
 + sqrt(x)*sqrt(d))/sqrt(c))*a*c**2*d**3*x + 6930*sqrt(d)*sqrt(c + d*x)*lo 
g((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b*c**5 + 6930*sqrt(d)*sqrt(c 
+ d*x)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b*c**4*d*x + 700*sqr 
t(d)*sqrt(c + d*x)*a*c**3*d**2 + 700*sqrt(d)*sqrt(c + d*x)*a*c**2*d**3*x + 
 1617*sqrt(d)*sqrt(c + d*x)*b*c**5 + 1617*sqrt(d)*sqrt(c + d*x)*b*c**4*d*x 
 - 3360*sqrt(x)*a*c**3*d**3 - 4480*sqrt(x)*a*c**2*d**4*x - 672*sqrt(x)*a*c 
*d**5*x**2 + 192*sqrt(x)*a*d**6*x**3 - 6930*sqrt(x)*b*c**5*d - 9240*sqrt(x 
)*b*c**4*d**2*x - 1386*sqrt(x)*b*c**3*d**3*x**2 + 396*sqrt(x)*b*c**2*d**4* 
x**3 - 176*sqrt(x)*b*c*d**5*x**4 + 96*sqrt(x)*b*d**6*x**5))/(384*sqrt(c + 
d*x)*d**7*(c + d*x))