Integrand size = 28, antiderivative size = 280 \[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}-\frac {2 (4 b c-3 a d)}{9 a^2 \sqrt {e x} \left (a x^2+b x^3\right )^{3/2}}-\frac {20 (4 b c-3 a d) e^2}{9 a^3 (e x)^{5/2} \sqrt {a x^2+b x^3}}+\frac {160 (4 b c-3 a d) e^4 \sqrt {a x^2+b x^3}}{63 a^4 (e x)^{9/2}}-\frac {64 b (4 b c-3 a d) e^3 \sqrt {a x^2+b x^3}}{21 a^5 (e x)^{7/2}}+\frac {256 b^2 (4 b c-3 a d) e^2 \sqrt {a x^2+b x^3}}{63 a^6 (e x)^{5/2}}-\frac {512 b^3 (4 b c-3 a d) e \sqrt {a x^2+b x^3}}{63 a^7 (e x)^{3/2}} \] Output:
-2/9*c*e/a/(e*x)^(3/2)/(b*x^3+a*x^2)^(3/2)-2/9*(-3*a*d+4*b*c)/a^2/(e*x)^(1 /2)/(b*x^3+a*x^2)^(3/2)-20/9*(-3*a*d+4*b*c)*e^2/a^3/(e*x)^(5/2)/(b*x^3+a*x ^2)^(1/2)+160/63*(-3*a*d+4*b*c)*e^4*(b*x^3+a*x^2)^(1/2)/a^4/(e*x)^(9/2)-64 /21*b*(-3*a*d+4*b*c)*e^3*(b*x^3+a*x^2)^(1/2)/a^5/(e*x)^(7/2)+256/63*b^2*(- 3*a*d+4*b*c)*e^2*(b*x^3+a*x^2)^(1/2)/a^6/(e*x)^(5/2)-512/63*b^3*(-3*a*d+4* b*c)*e*(b*x^3+a*x^2)^(1/2)/a^7/(e*x)^(3/2)
Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.48 \[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=-\frac {2 e \left (1024 b^6 c x^6+384 a^2 b^4 x^4 (c-3 d x)-768 a b^5 x^5 (-2 c+d x)+24 a^4 b^2 x^2 (c+2 d x)-6 a^5 b x (2 c+3 d x)-32 a^3 b^3 x^3 (2 c+9 d x)+a^6 (7 c+9 d x)\right )}{63 a^7 (e x)^{3/2} \left (x^2 (a+b x)\right )^{3/2}} \] Input:
Integrate[(c + d*x)/(Sqrt[e*x]*(a*x^2 + b*x^3)^(5/2)),x]
Output:
(-2*e*(1024*b^6*c*x^6 + 384*a^2*b^4*x^4*(c - 3*d*x) - 768*a*b^5*x^5*(-2*c + d*x) + 24*a^4*b^2*x^2*(c + 2*d*x) - 6*a^5*b*x*(2*c + 3*d*x) - 32*a^3*b^3 *x^3*(2*c + 9*d*x) + a^6*(7*c + 9*d*x)))/(63*a^7*(e*x)^(3/2)*(x^2*(a + b*x ))^(3/2))
Time = 0.87 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1944, 1921, 1921, 1922, 1922, 1922, 1920}
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.
| \(\displaystyle \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1944 |
| \(\displaystyle -\frac {(4 b c-3 a d) \int \frac {\sqrt {e x}}{\left (b x^3+a x^2\right )^{5/2}}dx}{3 a e}-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1921 |
| \(\displaystyle -\frac {(4 b c-3 a d) \left (\frac {10 e^2 \int \frac {1}{(e x)^{3/2} \left (b x^3+a x^2\right )^{3/2}}dx}{3 a}+\frac {2 e}{3 a \sqrt {e x} \left (a x^2+b x^3\right )^{3/2}}\right )}{3 a e}-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1921 |
| \(\displaystyle -\frac {(4 b c-3 a d) \left (\frac {10 e^2 \left (\frac {8 e^2 \int \frac {1}{(e x)^{7/2} \sqrt {b x^3+a x^2}}dx}{a}+\frac {2 e}{a (e x)^{5/2} \sqrt {a x^2+b x^3}}\right )}{3 a}+\frac {2 e}{3 a \sqrt {e x} \left (a x^2+b x^3\right )^{3/2}}\right )}{3 a e}-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {(4 b c-3 a d) \left (\frac {10 e^2 \left (\frac {8 e^2 \left (-\frac {6 b \int \frac {1}{(e x)^{5/2} \sqrt {b x^3+a x^2}}dx}{7 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{7 a (e x)^{9/2}}\right )}{a}+\frac {2 e}{a (e x)^{5/2} \sqrt {a x^2+b x^3}}\right )}{3 a}+\frac {2 e}{3 a \sqrt {e x} \left (a x^2+b x^3\right )^{3/2}}\right )}{3 a e}-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {(4 b c-3 a d) \left (\frac {10 e^2 \left (\frac {8 e^2 \left (-\frac {6 b \left (-\frac {4 b \int \frac {1}{(e x)^{3/2} \sqrt {b x^3+a x^2}}dx}{5 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{5 a (e x)^{7/2}}\right )}{7 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{7 a (e x)^{9/2}}\right )}{a}+\frac {2 e}{a (e x)^{5/2} \sqrt {a x^2+b x^3}}\right )}{3 a}+\frac {2 e}{3 a \sqrt {e x} \left (a x^2+b x^3\right )^{3/2}}\right )}{3 a e}-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {(4 b c-3 a d) \left (\frac {10 e^2 \left (\frac {8 e^2 \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {1}{\sqrt {e x} \sqrt {b x^3+a x^2}}dx}{3 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{3 a (e x)^{5/2}}\right )}{5 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{5 a (e x)^{7/2}}\right )}{7 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{7 a (e x)^{9/2}}\right )}{a}+\frac {2 e}{a (e x)^{5/2} \sqrt {a x^2+b x^3}}\right )}{3 a}+\frac {2 e}{3 a \sqrt {e x} \left (a x^2+b x^3\right )^{3/2}}\right )}{3 a e}-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle -\frac {(4 b c-3 a d) \left (\frac {10 e^2 \left (\frac {8 e^2 \left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b \sqrt {a x^2+b x^3}}{3 a^2 (e x)^{3/2}}-\frac {2 e \sqrt {a x^2+b x^3}}{3 a (e x)^{5/2}}\right )}{5 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{5 a (e x)^{7/2}}\right )}{7 a e}-\frac {2 e \sqrt {a x^2+b x^3}}{7 a (e x)^{9/2}}\right )}{a}+\frac {2 e}{a (e x)^{5/2} \sqrt {a x^2+b x^3}}\right )}{3 a}+\frac {2 e}{3 a \sqrt {e x} \left (a x^2+b x^3\right )^{3/2}}\right )}{3 a e}-\frac {2 c e}{9 a (e x)^{3/2} \left (a x^2+b x^3\right )^{3/2}}\) |
Input:
Int[(c + d*x)/(Sqrt[e*x]*(a*x^2 + b*x^3)^(5/2)),x]
Output:
(-2*c*e)/(9*a*(e*x)^(3/2)*(a*x^2 + b*x^3)^(3/2)) - ((4*b*c - 3*a*d)*((2*e) /(3*a*Sqrt[e*x]*(a*x^2 + b*x^3)^(3/2)) + (10*e^2*((2*e)/(a*(e*x)^(5/2)*Sqr t[a*x^2 + b*x^3]) + (8*e^2*((-2*e*Sqrt[a*x^2 + b*x^3])/(7*a*(e*x)^(9/2)) - (6*b*((-2*e*Sqrt[a*x^2 + b*x^3])/(5*a*(e*x)^(7/2)) - (4*b*((-2*e*Sqrt[a*x ^2 + b*x^3])/(3*a*(e*x)^(5/2)) + (4*b*Sqrt[a*x^2 + b*x^3])/(3*a^2*(e*x)^(3 /2))))/(5*a*e)))/(7*a*e)))/a))/(3*a)))/(3*a*e)
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[ n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n} , x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/( n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1) /(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Simp[(a*d*(m + j*p + 1) - b *c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)) Int[(e*x)^(m + n)*(a*x^ j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1 ] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && ( GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1 , 0]
Time = 0.47 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.58
| method | result | size |
| gosper | \(-\frac {2 x \left (b x +a \right ) \left (-768 a \,b^{5} d \,x^{6}+1024 b^{6} c \,x^{6}-1152 a^{2} b^{4} d \,x^{5}+1536 a \,b^{5} c \,x^{5}-288 a^{3} b^{3} d \,x^{4}+384 a^{2} b^{4} c \,x^{4}+48 a^{4} b^{2} d \,x^{3}-64 a^{3} b^{3} c \,x^{3}-18 a^{5} b d \,x^{2}+24 a^{4} b^{2} c \,x^{2}+9 a^{6} d x -12 a^{5} b c x +7 a^{6} c \right )}{63 a^{7} \sqrt {e x}\, \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}\) | \(163\) |
| default | \(-\frac {2 x \left (b x +a \right ) \left (-768 a \,b^{5} d \,x^{6}+1024 b^{6} c \,x^{6}-1152 a^{2} b^{4} d \,x^{5}+1536 a \,b^{5} c \,x^{5}-288 a^{3} b^{3} d \,x^{4}+384 a^{2} b^{4} c \,x^{4}+48 a^{4} b^{2} d \,x^{3}-64 a^{3} b^{3} c \,x^{3}-18 a^{5} b d \,x^{2}+24 a^{4} b^{2} c \,x^{2}+9 a^{6} d x -12 a^{5} b c x +7 a^{6} c \right )}{63 a^{7} \sqrt {e x}\, \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}\) | \(163\) |
| orering | \(-\frac {2 x \left (b x +a \right ) \left (-768 a \,b^{5} d \,x^{6}+1024 b^{6} c \,x^{6}-1152 a^{2} b^{4} d \,x^{5}+1536 a \,b^{5} c \,x^{5}-288 a^{3} b^{3} d \,x^{4}+384 a^{2} b^{4} c \,x^{4}+48 a^{4} b^{2} d \,x^{3}-64 a^{3} b^{3} c \,x^{3}-18 a^{5} b d \,x^{2}+24 a^{4} b^{2} c \,x^{2}+9 a^{6} d x -12 a^{5} b c x +7 a^{6} c \right )}{63 a^{7} \sqrt {e x}\, \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}\) | \(163\) |
| risch | \(-\frac {2 \left (b x +a \right ) \left (-474 x^{4} a \,b^{3} d +667 x^{4} b^{4} c +111 a^{2} b^{2} d \,x^{3}-176 a \,b^{3} c \,x^{3}-36 a^{3} b d \,x^{2}+69 a^{2} b^{2} c \,x^{2}+9 a^{4} d x -26 a^{3} b c x +7 c \,a^{4}\right )}{63 a^{7} x^{3} \sqrt {x^{2} \left (b x +a \right )}\, \sqrt {e x}}+\frac {2 b^{4} \left (14 a b d x -17 b^{2} c x +15 a^{2} d -18 a b c \right ) x^{2}}{3 \left (b x +a \right ) a^{7} \sqrt {x^{2} \left (b x +a \right )}\, \sqrt {e x}}\) | \(175\) |
Input:
int((d*x+c)/(e*x)^(1/2)/(b*x^3+a*x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/63*x*(b*x+a)*(-768*a*b^5*d*x^6+1024*b^6*c*x^6-1152*a^2*b^4*d*x^5+1536*a *b^5*c*x^5-288*a^3*b^3*d*x^4+384*a^2*b^4*c*x^4+48*a^4*b^2*d*x^3-64*a^3*b^3 *c*x^3-18*a^5*b*d*x^2+24*a^4*b^2*c*x^2+9*a^6*d*x-12*a^5*b*c*x+7*a^6*c)/a^7 /(e*x)^(1/2)/(b*x^3+a*x^2)^(5/2)
Time = 0.10 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.67 \[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=-\frac {2 \, {\left (7 \, a^{6} c + 256 \, {\left (4 \, b^{6} c - 3 \, a b^{5} d\right )} x^{6} + 384 \, {\left (4 \, a b^{5} c - 3 \, a^{2} b^{4} d\right )} x^{5} + 96 \, {\left (4 \, a^{2} b^{4} c - 3 \, a^{3} b^{3} d\right )} x^{4} - 16 \, {\left (4 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d\right )} x^{3} + 6 \, {\left (4 \, a^{4} b^{2} c - 3 \, a^{5} b d\right )} x^{2} - 3 \, {\left (4 \, a^{5} b c - 3 \, a^{6} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}} \sqrt {e x}}{63 \, {\left (a^{7} b^{2} e x^{8} + 2 \, a^{8} b e x^{7} + a^{9} e x^{6}\right )}} \] Input:
integrate((d*x+c)/(e*x)^(1/2)/(b*x^3+a*x^2)^(5/2),x, algorithm="fricas")
Output:
-2/63*(7*a^6*c + 256*(4*b^6*c - 3*a*b^5*d)*x^6 + 384*(4*a*b^5*c - 3*a^2*b^ 4*d)*x^5 + 96*(4*a^2*b^4*c - 3*a^3*b^3*d)*x^4 - 16*(4*a^3*b^3*c - 3*a^4*b^ 2*d)*x^3 + 6*(4*a^4*b^2*c - 3*a^5*b*d)*x^2 - 3*(4*a^5*b*c - 3*a^6*d)*x)*sq rt(b*x^3 + a*x^2)*sqrt(e*x)/(a^7*b^2*e*x^8 + 2*a^8*b*e*x^7 + a^9*e*x^6)
\[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=\int \frac {c + d x}{\sqrt {e x} \left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x+c)/(e*x)**(1/2)/(b*x**3+a*x**2)**(5/2),x)
Output:
Integral((c + d*x)/(sqrt(e*x)*(x**2*(a + b*x))**(5/2)), x)
\[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=\int { \frac {d x + c}{{\left (b x^{3} + a x^{2}\right )}^{\frac {5}{2}} \sqrt {e x}} \,d x } \] Input:
integrate((d*x+c)/(e*x)^(1/2)/(b*x^3+a*x^2)^(5/2),x, algorithm="maxima")
Output:
integrate((d*x + c)/((b*x^3 + a*x^2)^(5/2)*sqrt(e*x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (238) = 476\).
Time = 2.07 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.90 \[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (667 \, a^{18} b^{14} c e^{4} - 474 \, a^{19} b^{13} d e^{4}\right )} {\left (b x + a\right )}}{a^{25} b^{4} {\left | b \right |} \mathrm {sgn}\left (x\right )} - \frac {9 \, {\left (316 \, a^{19} b^{14} c e^{4} - 223 \, a^{20} b^{13} d e^{4}\right )}}{a^{25} b^{4} {\left | b \right |} \mathrm {sgn}\left (x\right )}\right )} + \frac {63 \, {\left (73 \, a^{20} b^{14} c e^{4} - 51 \, a^{21} b^{13} d e^{4}\right )}}{a^{25} b^{4} {\left | b \right |} \mathrm {sgn}\left (x\right )}\right )} - \frac {210 \, {\left (16 \, a^{21} b^{14} c e^{4} - 11 \, a^{22} b^{13} d e^{4}\right )}}{a^{25} b^{4} {\left | b \right |} \mathrm {sgn}\left (x\right )}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (3 \, a^{22} b^{14} c e^{4} - 2 \, a^{23} b^{13} d e^{4}\right )}}{a^{25} b^{4} {\left | b \right |} \mathrm {sgn}\left (x\right )}\right )} \sqrt {b x + a}}{63 \, {\left ({\left (b x + a\right )} b e - a b e\right )}^{\frac {9}{2}}} - \frac {4 \, {\left (17 \, \sqrt {b e} a^{2} b^{8} c e^{2} - 14 \, \sqrt {b e} a^{3} b^{7} d e^{2} + 36 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {{\left (b x + a\right )} b e - a b e}\right )}^{2} a b^{7} c e - 30 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {{\left (b x + a\right )} b e - a b e}\right )}^{2} a^{2} b^{6} d e + 15 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {{\left (b x + a\right )} b e - a b e}\right )}^{4} b^{6} c - 12 \, \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {{\left (b x + a\right )} b e - a b e}\right )}^{4} a b^{5} d\right )}}{3 \, {\left (a b e + {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3} a^{6} {\left | b \right |} \mathrm {sgn}\left (x\right )} \] Input:
integrate((d*x+c)/(e*x)^(1/2)/(b*x^3+a*x^2)^(5/2),x, algorithm="giac")
Output:
-2/63*(((b*x + a)*((b*x + a)*((667*a^18*b^14*c*e^4 - 474*a^19*b^13*d*e^4)* (b*x + a)/(a^25*b^4*abs(b)*sgn(x)) - 9*(316*a^19*b^14*c*e^4 - 223*a^20*b^1 3*d*e^4)/(a^25*b^4*abs(b)*sgn(x))) + 63*(73*a^20*b^14*c*e^4 - 51*a^21*b^13 *d*e^4)/(a^25*b^4*abs(b)*sgn(x))) - 210*(16*a^21*b^14*c*e^4 - 11*a^22*b^13 *d*e^4)/(a^25*b^4*abs(b)*sgn(x)))*(b*x + a) + 315*(3*a^22*b^14*c*e^4 - 2*a ^23*b^13*d*e^4)/(a^25*b^4*abs(b)*sgn(x)))*sqrt(b*x + a)/((b*x + a)*b*e - a *b*e)^(9/2) - 4/3*(17*sqrt(b*e)*a^2*b^8*c*e^2 - 14*sqrt(b*e)*a^3*b^7*d*e^2 + 36*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt((b*x + a)*b*e - a*b*e))^2* a*b^7*c*e - 30*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt((b*x + a)*b*e - a *b*e))^2*a^2*b^6*d*e + 15*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt((b*x + a)*b*e - a*b*e))^4*b^6*c - 12*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(( b*x + a)*b*e - a*b*e))^4*a*b^5*d)/((a*b*e + (sqrt(b*e)*sqrt(b*x + a) - sqr t((b*x + a)*b*e - a*b*e))^2)^3*a^6*abs(b)*sgn(x))
Time = 9.37 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.65 \[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {64\,b\,x^4\,\left (3\,a\,d-4\,b\,c\right )}{21\,a^5}-\frac {32\,x^3\,\left (3\,a\,d-4\,b\,c\right )}{63\,a^4}-\frac {2\,c}{9\,a\,b^2}-\frac {x\,\left (18\,a^6\,d-24\,a^5\,b\,c\right )}{63\,a^7\,b^2}+\frac {4\,x^2\,\left (3\,a\,d-4\,b\,c\right )}{21\,a^3\,b}+\frac {256\,b^2\,x^5\,\left (3\,a\,d-4\,b\,c\right )}{21\,a^6}+\frac {512\,b^3\,x^6\,\left (3\,a\,d-4\,b\,c\right )}{63\,a^7}\right )}{x^7\,\sqrt {e\,x}+\frac {a^2\,x^5\,\sqrt {e\,x}}{b^2}+\frac {2\,a\,x^6\,\sqrt {e\,x}}{b}} \] Input:
int((c + d*x)/((e*x)^(1/2)*(a*x^2 + b*x^3)^(5/2)),x)
Output:
((a*x^2 + b*x^3)^(1/2)*((64*b*x^4*(3*a*d - 4*b*c))/(21*a^5) - (32*x^3*(3*a *d - 4*b*c))/(63*a^4) - (2*c)/(9*a*b^2) - (x*(18*a^6*d - 24*a^5*b*c))/(63* a^7*b^2) + (4*x^2*(3*a*d - 4*b*c))/(21*a^3*b) + (256*b^2*x^5*(3*a*d - 4*b* c))/(21*a^6) + (512*b^3*x^6*(3*a*d - 4*b*c))/(63*a^7)))/(x^7*(e*x)^(1/2) + (a^2*x^5*(e*x)^(1/2))/b^2 + (2*a*x^6*(e*x)^(1/2))/b)
Time = 0.22 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x}{\sqrt {e x} \left (a x^2+b x^3\right )^{5/2}} \, dx=\frac {2 \sqrt {e}\, \left (-768 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{3} d \,x^{5}+1024 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{4} c \,x^{5}-768 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{4} d \,x^{6}+1024 \sqrt {b}\, \sqrt {b x +a}\, b^{5} c \,x^{6}-7 \sqrt {x}\, a^{6} c -9 \sqrt {x}\, a^{6} d x +12 \sqrt {x}\, a^{5} b c x +18 \sqrt {x}\, a^{5} b d \,x^{2}-24 \sqrt {x}\, a^{4} b^{2} c \,x^{2}-48 \sqrt {x}\, a^{4} b^{2} d \,x^{3}+64 \sqrt {x}\, a^{3} b^{3} c \,x^{3}+288 \sqrt {x}\, a^{3} b^{3} d \,x^{4}-384 \sqrt {x}\, a^{2} b^{4} c \,x^{4}+1152 \sqrt {x}\, a^{2} b^{4} d \,x^{5}-1536 \sqrt {x}\, a \,b^{5} c \,x^{5}+768 \sqrt {x}\, a \,b^{5} d \,x^{6}-1024 \sqrt {x}\, b^{6} c \,x^{6}\right )}{63 \sqrt {b x +a}\, a^{7} e \,x^{5} \left (b x +a \right )} \] Input:
int((d*x+c)/(e*x)^(1/2)/(b*x^3+a*x^2)^(5/2),x)
Output:
(2*sqrt(e)*( - 768*sqrt(b)*sqrt(a + b*x)*a**2*b**3*d*x**5 + 1024*sqrt(b)*s qrt(a + b*x)*a*b**4*c*x**5 - 768*sqrt(b)*sqrt(a + b*x)*a*b**4*d*x**6 + 102 4*sqrt(b)*sqrt(a + b*x)*b**5*c*x**6 - 7*sqrt(x)*a**6*c - 9*sqrt(x)*a**6*d* x + 12*sqrt(x)*a**5*b*c*x + 18*sqrt(x)*a**5*b*d*x**2 - 24*sqrt(x)*a**4*b** 2*c*x**2 - 48*sqrt(x)*a**4*b**2*d*x**3 + 64*sqrt(x)*a**3*b**3*c*x**3 + 288 *sqrt(x)*a**3*b**3*d*x**4 - 384*sqrt(x)*a**2*b**4*c*x**4 + 1152*sqrt(x)*a* *2*b**4*d*x**5 - 1536*sqrt(x)*a*b**5*c*x**5 + 768*sqrt(x)*a*b**5*d*x**6 - 1024*sqrt(x)*b**6*c*x**6))/(63*sqrt(a + b*x)*a**7*e*x**5*(a + b*x))