\(\int \frac {(d+e x)^{7/2}}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [308]

Optimal result

Integrand size = 39, antiderivative size = 236 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^4 d^4 \sqrt {d+e x}}+\frac {2 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{c^4 d^4 (d+e x)^{3/2}}+\frac {6 e^2 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^4 d^4 (d+e x)^{5/2}}+\frac {2 e^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^4 d^4 (d+e x)^{7/2}} \] Output:

2*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/(e*x+d) 
^(1/2)+2*e*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^ 
4/(e*x+d)^(3/2)+6/5*e^2*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 
5/2)/c^4/d^4/(e*x+d)^(5/2)+2/7*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2) 
/c^4/d^4/(e*x+d)^(7/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.56 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (7 d+e x)-2 a c^2 d^2 e^2 \left (35 d^2+14 d e x+3 e^2 x^2\right )+c^3 d^3 \left (35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3\right )\right )}{35 c^4 d^4 \sqrt {d+e x}} \] Input:

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

Output:

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d*e^4*(7*d + e*x) 
- 2*a*c^2*d^2*e^2*(35*d^2 + 14*d*e*x + 3*e^2*x^2) + c^3*d^3*(35*d^3 + 35*d 
^2*e*x + 21*d*e^2*x^2 + 5*e^3*x^3)))/(35*c^4*d^4*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1128, 1128, 1128, 1122}

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.

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1122

\(\displaystyle \frac {2 (d+e x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d}+\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{5 d}\right )}{7 d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 1122

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), 
 x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
EqQ[m + p, 0]
 

rule 1128

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d 
 + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
 

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.64

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-5 c^{3} d^{3} e^{3} x^{3}+6 x^{2} a \,c^{2} d^{2} e^{4}-21 c^{3} d^{4} e^{2} x^{2}-8 x \,a^{2} c d \,e^{5}+28 x a \,c^{2} d^{3} e^{3}-35 c^{3} d^{5} e x +16 e^{6} a^{3}-56 d^{2} e^{4} a^{2} c +70 d^{4} e^{2} a \,c^{2}-35 d^{6} c^{3}\right )}{35 \sqrt {e x +d}\, d^{4} c^{4}}\)	\(150\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-5 c^{3} d^{3} e^{3} x^{3}+6 x^{2} a \,c^{2} d^{2} e^{4}-21 c^{3} d^{4} e^{2} x^{2}-8 x \,a^{2} c d \,e^{5}+28 x a \,c^{2} d^{3} e^{3}-35 c^{3} d^{5} e x +16 e^{6} a^{3}-56 d^{2} e^{4} a^{2} c +70 d^{4} e^{2} a \,c^{2}-35 d^{6} c^{3}\right ) \sqrt {e x +d}}{35 d^{4} c^{4} \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(168\)
orering	\(-\frac {2 \left (-5 c^{3} d^{3} e^{3} x^{3}+6 x^{2} a \,c^{2} d^{2} e^{4}-21 c^{3} d^{4} e^{2} x^{2}-8 x \,a^{2} c d \,e^{5}+28 x a \,c^{2} d^{3} e^{3}-35 c^{3} d^{5} e x +16 e^{6} a^{3}-56 d^{2} e^{4} a^{2} c +70 d^{4} e^{2} a \,c^{2}-35 d^{6} c^{3}\right ) \left (c d x +a e \right ) \sqrt {e x +d}}{35 d^{4} c^{4} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\)	\(169\)

Input:

int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURN 
VERBOSE)
 

Output:

-2/35/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(-5*c^3*d^3*e^3*x^3+6*a*c^ 
2*d^2*e^4*x^2-21*c^3*d^4*e^2*x^2-8*a^2*c*d*e^5*x+28*a*c^2*d^3*e^3*x-35*c^3 
*d^5*e*x+16*a^3*e^6-56*a^2*c*d^2*e^4+70*a*c^2*d^4*e^2-35*c^3*d^6)/d^4/c^4
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (5 \, c^{3} d^{3} e^{3} x^{3} + 35 \, c^{3} d^{6} - 70 \, a c^{2} d^{4} e^{2} + 56 \, a^{2} c d^{2} e^{4} - 16 \, a^{3} e^{6} + 3 \, {\left (7 \, c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (35 \, c^{3} d^{5} e - 28 \, a c^{2} d^{3} e^{3} + 8 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{35 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \] Input:

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit 
hm="fricas")
 

Output:

2/35*(5*c^3*d^3*e^3*x^3 + 35*c^3*d^6 - 70*a*c^2*d^4*e^2 + 56*a^2*c*d^2*e^4 
 - 16*a^3*e^6 + 3*(7*c^3*d^4*e^2 - 2*a*c^2*d^2*e^4)*x^2 + (35*c^3*d^5*e - 
28*a*c^2*d^3*e^3 + 8*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e 
^2)*x)*sqrt(e*x + d)/(c^4*d^4*e*x + c^4*d^5)
 

Sympy [F(-1)]

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

integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (5 \, c^{4} d^{4} e^{3} x^{4} + 35 \, a c^{3} d^{6} e - 70 \, a^{2} c^{2} d^{4} e^{3} + 56 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + {\left (21 \, c^{4} d^{5} e^{2} - a c^{3} d^{3} e^{4}\right )} x^{3} + {\left (35 \, c^{4} d^{6} e - 7 \, a c^{3} d^{4} e^{3} + 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (35 \, c^{4} d^{7} - 35 \, a c^{3} d^{5} e^{2} + 28 \, a^{2} c^{2} d^{3} e^{4} - 8 \, a^{3} c d e^{6}\right )} x\right )}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} \] Input:

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit 
hm="maxima")
 

Output:

2/35*(5*c^4*d^4*e^3*x^4 + 35*a*c^3*d^6*e - 70*a^2*c^2*d^4*e^3 + 56*a^3*c*d 
^2*e^5 - 16*a^4*e^7 + (21*c^4*d^5*e^2 - a*c^3*d^3*e^4)*x^3 + (35*c^4*d^6*e 
 - 7*a*c^3*d^4*e^3 + 2*a^2*c^2*d^2*e^5)*x^2 + (35*c^4*d^7 - 35*a*c^3*d^5*e 
^2 + 28*a^2*c^2*d^3*e^4 - 8*a^3*c*d*e^6)*x)/(sqrt(c*d*x + a*e)*c^4*d^4)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, e {\left (\frac {35 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{c^{4} d^{4} e} + \frac {35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{4} e^{2} - 70 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d^{2} e^{4} + 35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} + 21 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d^{2} e - 21 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}}{c^{4} d^{4} e^{4}}\right )}}{35 \, {\left | e \right |}} \] Input:

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorit 
hm="giac")
 

Output:

2/35*e*(35*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt((e 
*x + d)*c*d*e - c*d^2*e + a*e^3)/(c^4*d^4*e) + (35*((e*x + d)*c*d*e - c*d^ 
2*e + a*e^3)^(3/2)*c^2*d^4*e^2 - 70*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3 
/2)*a*c*d^2*e^4 + 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 + 2 
1*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c*d^2*e - 21*((e*x + d)*c*d*e 
- c*d^2*e + a*e^3)^(5/2)*a*e^3 + 5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/ 
2))/(c^4*d^4*e^4))/abs(e)
 

Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (32\,a^3\,e^6-112\,a^2\,c\,d^2\,e^4+140\,a\,c^2\,d^4\,e^2-70\,c^3\,d^6\right )}{35\,c^4\,d^4\,e}-\frac {2\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^4-28\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{35\,c^3\,d^3}-\frac {2\,e^2\,x^3\,\sqrt {d+e\,x}}{7\,c\,d}+\frac {6\,e\,x^2\,\left (2\,a\,e^2-7\,c\,d^2\right )\,\sqrt {d+e\,x}}{35\,c^2\,d^2}\right )}{x+\frac {d}{e}} \] Input:

int((d + e*x)^(7/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)
 

Output:

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(((d + e*x)^(1/2)*(32*a^3* 
e^6 - 70*c^3*d^6 + 140*a*c^2*d^4*e^2 - 112*a^2*c*d^2*e^4))/(35*c^4*d^4*e) 
- (2*x*(d + e*x)^(1/2)*(8*a^2*e^4 + 35*c^2*d^4 - 28*a*c*d^2*e^2))/(35*c^3* 
d^3) - (2*e^2*x^3*(d + e*x)^(1/2))/(7*c*d) + (6*e*x^2*(2*a*e^2 - 7*c*d^2)* 
(d + e*x)^(1/2))/(35*c^2*d^2)))/(x + d/e)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.57 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (5 c^{3} d^{3} e^{3} x^{3}-6 a \,c^{2} d^{2} e^{4} x^{2}+21 c^{3} d^{4} e^{2} x^{2}+8 a^{2} c d \,e^{5} x -28 a \,c^{2} d^{3} e^{3} x +35 c^{3} d^{5} e x -16 a^{3} e^{6}+56 a^{2} c \,d^{2} e^{4}-70 a \,c^{2} d^{4} e^{2}+35 c^{3} d^{6}\right )}{35 c^{4} d^{4}} \] Input:

int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
 

Output:

(2*sqrt(a*e + c*d*x)*( - 16*a**3*e**6 + 56*a**2*c*d**2*e**4 + 8*a**2*c*d*e 
**5*x - 70*a*c**2*d**4*e**2 - 28*a*c**2*d**3*e**3*x - 6*a*c**2*d**2*e**4*x 
**2 + 35*c**3*d**6 + 35*c**3*d**5*e*x + 21*c**3*d**4*e**2*x**2 + 5*c**3*d* 
*3*e**3*x**3))/(35*c**4*d**4)