\(\int \frac {(d x)^{7/2}}{(a x+b x^2+c x^3)^{7/2}} \, dx\) [1]

Optimal result

Integrand size = 26, antiderivative size = 147 \[ \int \frac {(d x)^{7/2}}{\left (a x+b x^2+c x^3\right )^{7/2}} \, dx=-\frac {2 d (d x)^{5/2} (b+2 c x)}{5 \left (b^2-4 a c\right ) \left (a x+b x^2+c x^3\right )^{5/2}}+\frac {32 c d^2 (d x)^{3/2} (b+2 c x)}{15 \left (b^2-4 a c\right )^2 \left (a x+b x^2+c x^3\right )^{3/2}}-\frac {256 c^2 d^3 \sqrt {d x} (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a x+b x^2+c x^3}} \] Output:

-2/5*d*(d*x)^(5/2)*(2*c*x+b)/(-4*a*c+b^2)/(c*x^3+b*x^2+a*x)^(5/2)+32/15*c* 
d^2*(d*x)^(3/2)*(2*c*x+b)/(-4*a*c+b^2)^2/(c*x^3+b*x^2+a*x)^(3/2)-256/15*c^ 
2*d^3*(d*x)^(1/2)*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^3+b*x^2+a*x)^(1/2)
 

Mathematica [A] (verified)

Time = 2.40 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int \frac {(d x)^{7/2}}{\left (a x+b x^2+c x^3\right )^{7/2}} \, dx=-\frac {2 (d x)^{7/2} (b+2 c x) \left (a+b x+c x^2\right ) \left (3 b^4-40 a b^2 c+240 a^2 c^2-16 b^3 c x+320 a b c^2 x+112 b^2 c^2 x^2+320 a c^3 x^2+256 b c^3 x^3+128 c^4 x^4\right )}{15 \left (b^2-4 a c\right )^3 (x (a+x (b+c x)))^{7/2}} \] Input:

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

Output:

(-2*(d*x)^(7/2)*(b + 2*c*x)*(a + b*x + c*x^2)*(3*b^4 - 40*a*b^2*c + 240*a^ 
2*c^2 - 16*b^3*c*x + 320*a*b*c^2*x + 112*b^2*c^2*x^2 + 320*a*c^3*x^2 + 256 
*b*c^3*x^3 + 128*c^4*x^4))/(15*(b^2 - 4*a*c)^3*(x*(a + x*(b + c*x)))^(7/2) 
)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2467, 30, 1089, 1089, 1088}

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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 1089

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

\(\Big \downarrow \) 1089

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

\(\Big \downarrow \) 1088

\(\displaystyle \frac {d^3 \sqrt {d x} \sqrt {a+b x+c x^2} \left (-\frac {2 (b+2 c x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 c \left (\frac {16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}\right )}{\sqrt {a x+b x^2+c x^3}}\)

Input:

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

Output:

(d^3*Sqrt[d*x]*Sqrt[a + b*x + c*x^2]*((-2*(b + 2*c*x))/(5*(b^2 - 4*a*c)*(a 
 + b*x + c*x^2)^(5/2)) - (16*c*((-2*(b + 2*c*x))/(3*(b^2 - 4*a*c)*(a + b*x 
 + c*x^2)^(3/2)) + (16*c*(b + 2*c*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c* 
x^2])))/(5*(b^2 - 4*a*c))))/Sqrt[a*x + b*x^2 + c*x^3]
 

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 1088

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 
2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0]
 

rule 1089

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.18

method	result	size
gosper	\(\frac {2 \left (c \,x^{2}+b x +a \right ) \left (256 c^{5} x^{5}+640 b \,c^{4} x^{4}+640 a \,c^{4} x^{3}+480 b^{2} c^{3} x^{3}+960 a b \,c^{3} x^{2}+80 b^{3} c^{2} x^{2}+480 a^{2} c^{3} x +240 a \,b^{2} c^{2} x -10 b^{4} c x +240 a^{2} b \,c^{2}-40 a \,b^{3} c +3 b^{5}\right ) \left (d x \right )^{\frac {7}{2}}}{15 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (c \,x^{3}+b \,x^{2}+x a \right )^{\frac {7}{2}}}\)	\(174\)
orering	\(\frac {2 \left (c \,x^{2}+b x +a \right ) \left (256 c^{5} x^{5}+640 b \,c^{4} x^{4}+640 a \,c^{4} x^{3}+480 b^{2} c^{3} x^{3}+960 a b \,c^{3} x^{2}+80 b^{3} c^{2} x^{2}+480 a^{2} c^{3} x +240 a \,b^{2} c^{2} x -10 b^{4} c x +240 a^{2} b \,c^{2}-40 a \,b^{3} c +3 b^{5}\right ) \left (d x \right )^{\frac {7}{2}}}{15 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (c \,x^{3}+b \,x^{2}+x a \right )^{\frac {7}{2}}}\)	\(174\)
default	\(\frac {64 d^{3} \sqrt {d x}\, \sqrt {x \left (c \,x^{2}+b x +a \right )}\, \left (256 c^{5} x^{5}+640 b \,c^{4} x^{4}+640 a \,c^{4} x^{3}+480 b^{2} c^{3} x^{3}+960 a b \,c^{3} x^{2}+80 b^{3} c^{2} x^{2}+480 a^{2} c^{3} x +240 a \,b^{2} c^{2} x -10 b^{4} c x +240 a^{2} b \,c^{2}-40 a \,b^{3} c +3 b^{5}\right ) c^{3} \sqrt {-\frac {d \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right ) \left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right )}{c}}}{15 x \left (2 c x +\sqrt {-4 a c +b^{2}}+b \right )^{3} \left (b -\sqrt {-4 a c +b^{2}}+2 c x \right )^{3} \left (4 a c -b^{2}\right )^{3} \sqrt {d \left (c \,x^{2}+b x +a \right )}}\)	\(243\)

Input:

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

Output:

2/15*(c*x^2+b*x+a)*(256*c^5*x^5+640*b*c^4*x^4+640*a*c^4*x^3+480*b^2*c^3*x^ 
3+960*a*b*c^3*x^2+80*b^3*c^2*x^2+480*a^2*c^3*x+240*a*b^2*c^2*x-10*b^4*c*x+ 
240*a^2*b*c^2-40*a*b^3*c+3*b^5)*(d*x)^(7/2)/(64*a^3*c^3-48*a^2*b^2*c^2+12* 
a*b^4*c-b^6)/(c*x^3+b*x^2+a*x)^(7/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (129) = 258\).

Time = 0.10 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.12 \[ \int \frac {(d x)^{7/2}}{\left (a x+b x^2+c x^3\right )^{7/2}} \, dx=-\frac {2 \, {\left (256 \, c^{5} d^{3} x^{5} + 640 \, b c^{4} d^{3} x^{4} + 160 \, {\left (3 \, b^{2} c^{3} + 4 \, a c^{4}\right )} d^{3} x^{3} + 80 \, {\left (b^{3} c^{2} + 12 \, a b c^{3}\right )} d^{3} x^{2} - 10 \, {\left (b^{4} c - 24 \, a b^{2} c^{2} - 48 \, a^{2} c^{3}\right )} d^{3} x + {\left (3 \, b^{5} - 40 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} d^{3}\right )} \sqrt {c x^{3} + b x^{2} + a x} \sqrt {d x}}{15 \, {\left ({\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{7} + 3 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{6} + 3 \, {\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} x^{5} + {\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} x^{4} + 3 \, {\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} x^{3} + 3 \, {\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} x^{2} + {\left (a^{3} b^{6} - 12 \, a^{4} b^{4} c + 48 \, a^{5} b^{2} c^{2} - 64 \, a^{6} c^{3}\right )} x\right )}} \] Input:

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

Output:

-2/15*(256*c^5*d^3*x^5 + 640*b*c^4*d^3*x^4 + 160*(3*b^2*c^3 + 4*a*c^4)*d^3 
*x^3 + 80*(b^3*c^2 + 12*a*b*c^3)*d^3*x^2 - 10*(b^4*c - 24*a*b^2*c^2 - 48*a 
^2*c^3)*d^3*x + (3*b^5 - 40*a*b^3*c + 240*a^2*b*c^2)*d^3)*sqrt(c*x^3 + b*x 
^2 + a*x)*sqrt(d*x)/((b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6 
)*x^7 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^6 + 3 
*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^5 
 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^ 
4 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4 
)*x^3 + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x^2 + ( 
a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*x)
 

Sympy [F]

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

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

Output:

Integral((d*x)**(7/2)/(x*(a + b*x + c*x**2))**(7/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (129) = 258\).

Time = 0.21 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.76 \[ \int \frac {(d x)^{7/2}}{\left (a x+b x^2+c x^3\right )^{7/2}} \, dx=-\frac {2}{15} \, {\left (\frac {{\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, c^{5} d^{2} x}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} + \frac {5 \, b c^{4} d^{2}}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}}\right )} d x + \frac {5 \, {\left (3 \, b^{2} c^{3} d^{3} + 4 \, a c^{4} d^{3}\right )}}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}}\right )} d x + \frac {5 \, {\left (b^{3} c^{2} d^{4} + 12 \, a b c^{3} d^{4}\right )}}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}}\right )} d x - \frac {5 \, {\left (b^{4} c d^{5} - 24 \, a b^{2} c^{2} d^{5} - 48 \, a^{2} c^{3} d^{5}\right )}}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}}\right )} d x + \frac {3 \, b^{5} d^{6} - 40 \, a b^{3} c d^{6} + 240 \, a^{2} b c^{2} d^{6}}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}}\right )} d^{8}}{{\left (c d^{3} x^{2} + b d^{3} x + a d^{3}\right )}^{\frac {5}{2}}} - \frac {3 \, b^{5} d^{3} - 40 \, a b^{3} c d^{3} + 240 \, a^{2} b c^{2} d^{3}}{\sqrt {a d^{3}} a^{2} b^{6} - 12 \, \sqrt {a d^{3}} a^{3} b^{4} c + 48 \, \sqrt {a d^{3}} a^{4} b^{2} c^{2} - 64 \, \sqrt {a d^{3}} a^{5} c^{3}}\right )} d^{2} \] Input:

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

Output:

-2/15*((2*(8*(2*(4*(2*c^5*d^2*x/(b^6*d^5 - 12*a*b^4*c*d^5 + 48*a^2*b^2*c^2 
*d^5 - 64*a^3*c^3*d^5) + 5*b*c^4*d^2/(b^6*d^5 - 12*a*b^4*c*d^5 + 48*a^2*b^ 
2*c^2*d^5 - 64*a^3*c^3*d^5))*d*x + 5*(3*b^2*c^3*d^3 + 4*a*c^4*d^3)/(b^6*d^ 
5 - 12*a*b^4*c*d^5 + 48*a^2*b^2*c^2*d^5 - 64*a^3*c^3*d^5))*d*x + 5*(b^3*c^ 
2*d^4 + 12*a*b*c^3*d^4)/(b^6*d^5 - 12*a*b^4*c*d^5 + 48*a^2*b^2*c^2*d^5 - 6 
4*a^3*c^3*d^5))*d*x - 5*(b^4*c*d^5 - 24*a*b^2*c^2*d^5 - 48*a^2*c^3*d^5)/(b 
^6*d^5 - 12*a*b^4*c*d^5 + 48*a^2*b^2*c^2*d^5 - 64*a^3*c^3*d^5))*d*x + (3*b 
^5*d^6 - 40*a*b^3*c*d^6 + 240*a^2*b*c^2*d^6)/(b^6*d^5 - 12*a*b^4*c*d^5 + 4 
8*a^2*b^2*c^2*d^5 - 64*a^3*c^3*d^5))*d^8/(c*d^3*x^2 + b*d^3*x + a*d^3)^(5/ 
2) - (3*b^5*d^3 - 40*a*b^3*c*d^3 + 240*a^2*b*c^2*d^3)/(sqrt(a*d^3)*a^2*b^6 
 - 12*sqrt(a*d^3)*a^3*b^4*c + 48*sqrt(a*d^3)*a^4*b^2*c^2 - 64*sqrt(a*d^3)* 
a^5*c^3))*d^2
 

Mupad [B] (verification not implemented)

Time = 21.98 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.18 \[ \int \frac {(d x)^{7/2}}{\left (a x+b x^2+c x^3\right )^{7/2}} \, dx=\frac {\sqrt {c\,x^3+b\,x^2+a\,x}\,\left (\frac {64\,d^3\,x^3\,\sqrt {d\,x}\,\left (3\,b^2+4\,a\,c\right )}{3\,{\left (4\,a\,c-b^2\right )}^3}+\frac {512\,c^2\,d^3\,x^5\,\sqrt {d\,x}}{15\,{\left (4\,a\,c-b^2\right )}^3}+\frac {2\,b\,d^3\,\sqrt {d\,x}\,\left (240\,a^2\,c^2-40\,a\,b^2\,c+3\,b^4\right )}{15\,c^3\,{\left (4\,a\,c-b^2\right )}^3}+\frac {4\,d^3\,x\,\sqrt {d\,x}\,\left (48\,a^2\,c^2+24\,a\,b^2\,c-b^4\right )}{3\,c^2\,{\left (4\,a\,c-b^2\right )}^3}+\frac {256\,b\,c\,d^3\,x^4\,\sqrt {d\,x}}{3\,{\left (4\,a\,c-b^2\right )}^3}+\frac {32\,b\,d^3\,x^2\,\sqrt {d\,x}\,\left (b^2+12\,a\,c\right )}{3\,c\,{\left (4\,a\,c-b^2\right )}^3}\right )}{x^7+\frac {3\,x^5\,\left (b^2+a\,c\right )}{c^2}+\frac {a^3\,x}{c^3}+\frac {3\,b\,x^6}{c}+\frac {3\,a^2\,b\,x^2}{c^3}+\frac {3\,a\,x^3\,\left (b^2+a\,c\right )}{c^3}+\frac {b\,x^4\,\left (b^2+6\,a\,c\right )}{c^3}} \] Input:

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

Output:

((a*x + b*x^2 + c*x^3)^(1/2)*((64*d^3*x^3*(d*x)^(1/2)*(4*a*c + 3*b^2))/(3* 
(4*a*c - b^2)^3) + (512*c^2*d^3*x^5*(d*x)^(1/2))/(15*(4*a*c - b^2)^3) + (2 
*b*d^3*(d*x)^(1/2)*(3*b^4 + 240*a^2*c^2 - 40*a*b^2*c))/(15*c^3*(4*a*c - b^ 
2)^3) + (4*d^3*x*(d*x)^(1/2)*(48*a^2*c^2 - b^4 + 24*a*b^2*c))/(3*c^2*(4*a* 
c - b^2)^3) + (256*b*c*d^3*x^4*(d*x)^(1/2))/(3*(4*a*c - b^2)^3) + (32*b*d^ 
3*x^2*(d*x)^(1/2)*(12*a*c + b^2))/(3*c*(4*a*c - b^2)^3)))/(x^7 + (3*x^5*(a 
*c + b^2))/c^2 + (a^3*x)/c^3 + (3*b*x^6)/c + (3*a^2*b*x^2)/c^3 + (3*a*x^3* 
(a*c + b^2))/c^3 + (b*x^4*(6*a*c + b^2))/c^3)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 719, normalized size of antiderivative = 4.89 \[ \int \frac {(d x)^{7/2}}{\left (a x+b x^2+c x^3\right )^{7/2}} \, dx=\frac {2 \sqrt {d}\, d^{3} \left (240 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{2}+480 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3} x -40 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c +240 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2} x +960 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} x^{2}+640 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} x^{3}+3 \sqrt {c \,x^{2}+b x +a}\, b^{5}-10 \sqrt {c \,x^{2}+b x +a}\, b^{4} c x +80 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{2} x^{2}+480 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} x^{3}+640 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} x^{4}+256 \sqrt {c \,x^{2}+b x +a}\, c^{5} x^{5}-256 \sqrt {c}\, a^{3} c^{2}-768 \sqrt {c}\, a^{2} b \,c^{2} x -768 \sqrt {c}\, a^{2} c^{3} x^{2}-768 \sqrt {c}\, a \,b^{2} c^{2} x^{2}-1536 \sqrt {c}\, a b \,c^{3} x^{3}-768 \sqrt {c}\, a \,c^{4} x^{4}-256 \sqrt {c}\, b^{3} c^{2} x^{3}-768 \sqrt {c}\, b^{2} c^{3} x^{4}-768 \sqrt {c}\, b \,c^{4} x^{5}-256 \sqrt {c}\, c^{5} x^{6}\right )}{960 a^{3} c^{6} x^{6}-720 a^{2} b^{2} c^{5} x^{6}+180 a \,b^{4} c^{4} x^{6}-15 b^{6} c^{3} x^{6}+2880 a^{3} b \,c^{5} x^{5}-2160 a^{2} b^{3} c^{4} x^{5}+540 a \,b^{5} c^{3} x^{5}-45 b^{7} c^{2} x^{5}+2880 a^{4} c^{5} x^{4}+720 a^{3} b^{2} c^{4} x^{4}-1620 a^{2} b^{4} c^{3} x^{4}+495 a \,b^{6} c^{2} x^{4}-45 b^{8} c \,x^{4}+5760 a^{4} b \,c^{4} x^{3}-3360 a^{3} b^{3} c^{3} x^{3}+360 a^{2} b^{5} c^{2} x^{3}+90 a \,b^{7} c \,x^{3}-15 b^{9} x^{3}+2880 a^{5} c^{4} x^{2}+720 a^{4} b^{2} c^{3} x^{2}-1620 a^{3} b^{4} c^{2} x^{2}+495 a^{2} b^{6} c \,x^{2}-45 a \,b^{8} x^{2}+2880 a^{5} b \,c^{3} x -2160 a^{4} b^{3} c^{2} x +540 a^{3} b^{5} c x -45 a^{2} b^{7} x +960 a^{6} c^{3}-720 a^{5} b^{2} c^{2}+180 a^{4} b^{4} c -15 a^{3} b^{6}} \] Input:

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

Output:

(2*sqrt(d)*d**3*(240*sqrt(a + b*x + c*x**2)*a**2*b*c**2 + 480*sqrt(a + b*x 
 + c*x**2)*a**2*c**3*x - 40*sqrt(a + b*x + c*x**2)*a*b**3*c + 240*sqrt(a + 
 b*x + c*x**2)*a*b**2*c**2*x + 960*sqrt(a + b*x + c*x**2)*a*b*c**3*x**2 + 
640*sqrt(a + b*x + c*x**2)*a*c**4*x**3 + 3*sqrt(a + b*x + c*x**2)*b**5 - 1 
0*sqrt(a + b*x + c*x**2)*b**4*c*x + 80*sqrt(a + b*x + c*x**2)*b**3*c**2*x* 
*2 + 480*sqrt(a + b*x + c*x**2)*b**2*c**3*x**3 + 640*sqrt(a + b*x + c*x**2 
)*b*c**4*x**4 + 256*sqrt(a + b*x + c*x**2)*c**5*x**5 - 256*sqrt(c)*a**3*c* 
*2 - 768*sqrt(c)*a**2*b*c**2*x - 768*sqrt(c)*a**2*c**3*x**2 - 768*sqrt(c)* 
a*b**2*c**2*x**2 - 1536*sqrt(c)*a*b*c**3*x**3 - 768*sqrt(c)*a*c**4*x**4 - 
256*sqrt(c)*b**3*c**2*x**3 - 768*sqrt(c)*b**2*c**3*x**4 - 768*sqrt(c)*b*c* 
*4*x**5 - 256*sqrt(c)*c**5*x**6))/(15*(64*a**6*c**3 - 48*a**5*b**2*c**2 + 
192*a**5*b*c**3*x + 192*a**5*c**4*x**2 + 12*a**4*b**4*c - 144*a**4*b**3*c* 
*2*x + 48*a**4*b**2*c**3*x**2 + 384*a**4*b*c**4*x**3 + 192*a**4*c**5*x**4 
- a**3*b**6 + 36*a**3*b**5*c*x - 108*a**3*b**4*c**2*x**2 - 224*a**3*b**3*c 
**3*x**3 + 48*a**3*b**2*c**4*x**4 + 192*a**3*b*c**5*x**5 + 64*a**3*c**6*x* 
*6 - 3*a**2*b**7*x + 33*a**2*b**6*c*x**2 + 24*a**2*b**5*c**2*x**3 - 108*a* 
*2*b**4*c**3*x**4 - 144*a**2*b**3*c**4*x**5 - 48*a**2*b**2*c**5*x**6 - 3*a 
*b**8*x**2 + 6*a*b**7*c*x**3 + 33*a*b**6*c**2*x**4 + 36*a*b**5*c**3*x**5 + 
 12*a*b**4*c**4*x**6 - b**9*x**3 - 3*b**8*c*x**4 - 3*b**7*c**2*x**5 - b**6 
*c**3*x**6))