Integrand size = 21, antiderivative size = 248 \[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\frac {2 b^3 (b c-a d) x^3}{3 a^5 \left (a x^2+b x^3\right )^{3/2}}+\frac {2 b^3 (5 b c-4 a d) x}{a^6 \sqrt {a x^2+b x^3}}-\frac {c \sqrt {a x^2+b x^3}}{4 a^3 x^5}+\frac {(23 b c-8 a d) \sqrt {a x^2+b x^3}}{24 a^4 x^4}-\frac {b (259 b c-136 a d) \sqrt {a x^2+b x^3}}{96 a^5 x^3}+\frac {b^2 (515 b c-328 a d) \sqrt {a x^2+b x^3}}{64 a^6 x^2}-\frac {105 b^3 (11 b c-8 a d) \text {arctanh}\left (\frac {\sqrt {a x^2+b x^3}}{\sqrt {a} x}\right )}{64 a^{13/2}} \] Output:
2/3*b^3*(-a*d+b*c)*x^3/a^5/(b*x^3+a*x^2)^(3/2)+2*b^3*(-4*a*d+5*b*c)*x/a^6/ (b*x^3+a*x^2)^(1/2)-1/4*c*(b*x^3+a*x^2)^(1/2)/a^3/x^5+1/24*(-8*a*d+23*b*c) *(b*x^3+a*x^2)^(1/2)/a^4/x^4-1/96*b*(-136*a*d+259*b*c)*(b*x^3+a*x^2)^(1/2) /a^5/x^3+1/64*b^2*(-328*a*d+515*b*c)*(b*x^3+a*x^2)^(1/2)/a^6/x^2-105/64*b^ 3*(-8*a*d+11*b*c)*arctanh((b*x^3+a*x^2)^(1/2)/a^(1/2)/x)/a^(13/2)
Time = 0.32 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.67 \[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\frac {\sqrt {a} \left (3465 b^5 c x^5+21 a^2 b^3 x^3 (33 c-160 d x)+420 a b^4 x^4 (11 c-6 d x)-16 a^5 (3 c+4 d x)+8 a^4 b x (11 c+18 d x)-18 a^3 b^2 x^2 (11 c+28 d x)\right )+315 b^3 (-11 b c+8 a d) x^4 (a+b x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{192 a^{13/2} x \left (x^2 (a+b x)\right )^{3/2}} \] Input:
Integrate[(c + d*x)/(a*x^2 + b*x^3)^(5/2),x]
Output:
(Sqrt[a]*(3465*b^5*c*x^5 + 21*a^2*b^3*x^3*(33*c - 160*d*x) + 420*a*b^4*x^4 *(11*c - 6*d*x) - 16*a^5*(3*c + 4*d*x) + 8*a^4*b*x*(11*c + 18*d*x) - 18*a^ 3*b^2*x^2*(11*c + 28*d*x)) + 315*b^3*(-11*b*c + 8*a*d)*x^4*(a + b*x)^(3/2) *ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(192*a^(13/2)*x*(x^2*(a + b*x))^(3/2))
Time = 0.93 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.46, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2450, 2009}
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.
| \(\displaystyle \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2450 |
| \(\displaystyle \int \left (\frac {c}{\left (a x^2+b x^3\right )^{5/2}}+\frac {d x}{\left (a x^2+b x^3\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1155 b^4 c \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{13/2}}+\frac {105 b^3 d \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{11/2}}+\frac {1155 b^3 c \sqrt {a x^2+b x^3}}{64 a^6 x^2}-\frac {385 b^2 c \sqrt {a x^2+b x^3}}{32 a^5 x^3}-\frac {105 b^2 d \sqrt {a x^2+b x^3}}{8 a^5 x^2}+\frac {77 b c \sqrt {a x^2+b x^3}}{8 a^4 x^4}+\frac {35 b d \sqrt {a x^2+b x^3}}{4 a^4 x^3}-\frac {33 c \sqrt {a x^2+b x^3}}{4 a^3 x^5}-\frac {7 d \sqrt {a x^2+b x^3}}{a^3 x^4}+\frac {22 c}{3 a^2 x^3 \sqrt {a x^2+b x^3}}+\frac {6 d}{a^2 x^2 \sqrt {a x^2+b x^3}}+\frac {2 c}{3 a x \left (a x^2+b x^3\right )^{3/2}}+\frac {2 d}{3 a \left (a x^2+b x^3\right )^{3/2}}\) |
Input:
Int[(c + d*x)/(a*x^2 + b*x^3)^(5/2),x]
Output:
(2*d)/(3*a*(a*x^2 + b*x^3)^(3/2)) + (2*c)/(3*a*x*(a*x^2 + b*x^3)^(3/2)) + (22*c)/(3*a^2*x^3*Sqrt[a*x^2 + b*x^3]) + (6*d)/(a^2*x^2*Sqrt[a*x^2 + b*x^3 ]) - (33*c*Sqrt[a*x^2 + b*x^3])/(4*a^3*x^5) + (77*b*c*Sqrt[a*x^2 + b*x^3]) /(8*a^4*x^4) - (7*d*Sqrt[a*x^2 + b*x^3])/(a^3*x^4) - (385*b^2*c*Sqrt[a*x^2 + b*x^3])/(32*a^5*x^3) + (35*b*d*Sqrt[a*x^2 + b*x^3])/(4*a^4*x^3) + (1155 *b^3*c*Sqrt[a*x^2 + b*x^3])/(64*a^6*x^2) - (105*b^2*d*Sqrt[a*x^2 + b*x^3]) /(8*a^5*x^2) - (1155*b^4*c*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(64*a ^(13/2)) + (105*b^3*d*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(8*a^(11/2 ))
Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[Expan dIntegrand[Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && (Po lyQ[Pq, x] || PolyQ[Pq, x^n]) && !IntegerQ[p] && NeQ[n, j]
Time = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.10
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (\frac {\left (3 d x +c \right ) b}{2}+a d \right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{2}}\) | \(26\) |
| risch | \(-\frac {\left (b x +a \right ) \left (984 a \,b^{2} d \,x^{3}-1545 b^{3} c \,x^{3}-272 a^{2} b d \,x^{2}+518 a \,b^{2} c \,x^{2}+64 a^{3} d x -184 a^{2} b c x +48 c \,a^{3}\right )}{192 a^{6} x^{3} \sqrt {x^{2} \left (b x +a \right )}}-\frac {b^{3} \left (-\frac {2 \left (840 a d -1155 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 \left (-512 a d +640 b c \right )}{\sqrt {b x +a}}+\frac {256 a \left (a d -b c \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}\right ) \sqrt {b x +a}\, x}{128 a^{6} \sqrt {x^{2} \left (b x +a \right )}}\) | \(177\) |
| default | \(\frac {x \left (b x +a \right ) \left (2520 \left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a \,b^{3} d \,x^{4}-3465 \left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{4} c \,x^{4}-64 a^{\frac {11}{2}} d x +144 a^{\frac {9}{2}} b d \,x^{2}-504 a^{\frac {7}{2}} b^{2} d \,x^{3}-3360 a^{\frac {5}{2}} b^{3} d \,x^{4}-2520 a^{\frac {3}{2}} b^{4} d \,x^{5}-48 a^{\frac {11}{2}} c +88 a^{\frac {9}{2}} b c x -198 a^{\frac {7}{2}} b^{2} c \,x^{2}+693 a^{\frac {5}{2}} b^{3} c \,x^{3}+4620 a^{\frac {3}{2}} b^{4} c \,x^{4}+3465 \sqrt {a}\, b^{5} c \,x^{5}\right )}{192 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}} a^{\frac {13}{2}}}\) | \(198\) |
Input:
int((d*x+c)/(b*x^3+a*x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-4/3/(b*x+a)^(3/2)*(1/2*(3*d*x+c)*b+a*d)/b^2
Time = 0.12 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.14 \[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\left [-\frac {315 \, {\left ({\left (11 \, b^{6} c - 8 \, a b^{5} d\right )} x^{7} + 2 \, {\left (11 \, a b^{5} c - 8 \, a^{2} b^{4} d\right )} x^{6} + {\left (11 \, a^{2} b^{4} c - 8 \, a^{3} b^{3} d\right )} x^{5}\right )} \sqrt {a} \log \left (\frac {b x^{2} + 2 \, a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (48 \, a^{6} c - 315 \, {\left (11 \, a b^{5} c - 8 \, a^{2} b^{4} d\right )} x^{5} - 420 \, {\left (11 \, a^{2} b^{4} c - 8 \, a^{3} b^{3} d\right )} x^{4} - 63 \, {\left (11 \, a^{3} b^{3} c - 8 \, a^{4} b^{2} d\right )} x^{3} + 18 \, {\left (11 \, a^{4} b^{2} c - 8 \, a^{5} b d\right )} x^{2} - 8 \, {\left (11 \, a^{5} b c - 8 \, a^{6} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{384 \, {\left (a^{7} b^{2} x^{7} + 2 \, a^{8} b x^{6} + a^{9} x^{5}\right )}}, \frac {315 \, {\left ({\left (11 \, b^{6} c - 8 \, a b^{5} d\right )} x^{7} + 2 \, {\left (11 \, a b^{5} c - 8 \, a^{2} b^{4} d\right )} x^{6} + {\left (11 \, a^{2} b^{4} c - 8 \, a^{3} b^{3} d\right )} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{b x^{2} + a x}\right ) - {\left (48 \, a^{6} c - 315 \, {\left (11 \, a b^{5} c - 8 \, a^{2} b^{4} d\right )} x^{5} - 420 \, {\left (11 \, a^{2} b^{4} c - 8 \, a^{3} b^{3} d\right )} x^{4} - 63 \, {\left (11 \, a^{3} b^{3} c - 8 \, a^{4} b^{2} d\right )} x^{3} + 18 \, {\left (11 \, a^{4} b^{2} c - 8 \, a^{5} b d\right )} x^{2} - 8 \, {\left (11 \, a^{5} b c - 8 \, a^{6} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{192 \, {\left (a^{7} b^{2} x^{7} + 2 \, a^{8} b x^{6} + a^{9} x^{5}\right )}}\right ] \] Input:
integrate((d*x+c)/(b*x^3+a*x^2)^(5/2),x, algorithm="fricas")
Output:
[-1/384*(315*((11*b^6*c - 8*a*b^5*d)*x^7 + 2*(11*a*b^5*c - 8*a^2*b^4*d)*x^ 6 + (11*a^2*b^4*c - 8*a^3*b^3*d)*x^5)*sqrt(a)*log((b*x^2 + 2*a*x + 2*sqrt( b*x^3 + a*x^2)*sqrt(a))/x^2) + 2*(48*a^6*c - 315*(11*a*b^5*c - 8*a^2*b^4*d )*x^5 - 420*(11*a^2*b^4*c - 8*a^3*b^3*d)*x^4 - 63*(11*a^3*b^3*c - 8*a^4*b^ 2*d)*x^3 + 18*(11*a^4*b^2*c - 8*a^5*b*d)*x^2 - 8*(11*a^5*b*c - 8*a^6*d)*x) *sqrt(b*x^3 + a*x^2))/(a^7*b^2*x^7 + 2*a^8*b*x^6 + a^9*x^5), 1/192*(315*(( 11*b^6*c - 8*a*b^5*d)*x^7 + 2*(11*a*b^5*c - 8*a^2*b^4*d)*x^6 + (11*a^2*b^4 *c - 8*a^3*b^3*d)*x^5)*sqrt(-a)*arctan(sqrt(b*x^3 + a*x^2)*sqrt(-a)/(b*x^2 + a*x)) - (48*a^6*c - 315*(11*a*b^5*c - 8*a^2*b^4*d)*x^5 - 420*(11*a^2*b^ 4*c - 8*a^3*b^3*d)*x^4 - 63*(11*a^3*b^3*c - 8*a^4*b^2*d)*x^3 + 18*(11*a^4* b^2*c - 8*a^5*b*d)*x^2 - 8*(11*a^5*b*c - 8*a^6*d)*x)*sqrt(b*x^3 + a*x^2))/ (a^7*b^2*x^7 + 2*a^8*b*x^6 + a^9*x^5)]
\[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\int \frac {c + d x}{\left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x+c)/(b*x**3+a*x**2)**(5/2),x)
Output:
Integral((c + d*x)/(x**2*(a + b*x))**(5/2), x)
\[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\int { \frac {d x + c}{{\left (b x^{3} + a x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)/(b*x^3+a*x^2)^(5/2),x, algorithm="maxima")
Output:
integrate((d*x + c)/(b*x^3 + a*x^2)^(5/2), x)
Time = 0.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\frac {105 \, {\left (11 \, b^{4} c - 8 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{6} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (15 \, {\left (b x + a\right )} b^{4} c + a b^{4} c - 12 \, {\left (b x + a\right )} a b^{3} d - a^{2} b^{3} d\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} \mathrm {sgn}\left (x\right )} + \frac {1545 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{4} c - 5153 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{4} c + 5855 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{4} c - 2295 \, \sqrt {b x + a} a^{3} b^{4} c - 984 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{3} d + 3224 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{3} d - 3560 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{3} d + 1320 \, \sqrt {b x + a} a^{4} b^{3} d}{192 \, a^{6} b^{4} x^{4} \mathrm {sgn}\left (x\right )} \] Input:
integrate((d*x+c)/(b*x^3+a*x^2)^(5/2),x, algorithm="giac")
Output:
105/64*(11*b^4*c - 8*a*b^3*d)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^6 *sgn(x)) + 2/3*(15*(b*x + a)*b^4*c + a*b^4*c - 12*(b*x + a)*a*b^3*d - a^2* b^3*d)/((b*x + a)^(3/2)*a^6*sgn(x)) + 1/192*(1545*(b*x + a)^(7/2)*b^4*c - 5153*(b*x + a)^(5/2)*a*b^4*c + 5855*(b*x + a)^(3/2)*a^2*b^4*c - 2295*sqrt( b*x + a)*a^3*b^4*c - 984*(b*x + a)^(7/2)*a*b^3*d + 3224*(b*x + a)^(5/2)*a^ 2*b^3*d - 3560*(b*x + a)^(3/2)*a^3*b^3*d + 1320*sqrt(b*x + a)*a^4*b^3*d)/( a^6*b^4*x^4*sgn(x))
Timed out. \[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\int \frac {c+d\,x}{{\left (b\,x^3+a\,x^2\right )}^{5/2}} \,d x \] Input:
int((c + d*x)/(a*x^2 + b*x^3)^(5/2),x)
Output:
int((c + d*x)/(a*x^2 + b*x^3)^(5/2), x)
Time = 0.24 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.50 \[ \int \frac {c+d x}{\left (a x^2+b x^3\right )^{5/2}} \, dx=\frac {-2520 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} b^{3} d \,x^{4}+3465 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{4} c \,x^{4}-2520 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{4} d \,x^{5}+3465 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{5} c \,x^{5}+2520 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} b^{3} d \,x^{4}-3465 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{4} c \,x^{4}+2520 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{4} d \,x^{5}-3465 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{5} c \,x^{5}-96 a^{6} c -128 a^{6} d x +176 a^{5} b c x +288 a^{5} b d \,x^{2}-396 a^{4} b^{2} c \,x^{2}-1008 a^{4} b^{2} d \,x^{3}+1386 a^{3} b^{3} c \,x^{3}-6720 a^{3} b^{3} d \,x^{4}+9240 a^{2} b^{4} c \,x^{4}-5040 a^{2} b^{4} d \,x^{5}+6930 a \,b^{5} c \,x^{5}}{384 \sqrt {b x +a}\, a^{7} x^{4} \left (b x +a \right )} \] Input:
int((d*x+c)/(b*x^3+a*x^2)^(5/2),x)
Output:
( - 2520*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*a**2*b**3*d*x* *4 + 3465*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*a*b**4*c*x**4 - 2520*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*a*b**4*d*x**5 + 3465*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*b**5*c*x**5 + 252 0*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a**2*b**3*d*x**4 - 34 65*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a*b**4*c*x**4 + 2520 *sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a*b**4*d*x**5 - 3465*s qrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*b**5*c*x**5 - 96*a**6*c - 128*a**6*d*x + 176*a**5*b*c*x + 288*a**5*b*d*x**2 - 396*a**4*b**2*c*x**2 - 1008*a**4*b**2*d*x**3 + 1386*a**3*b**3*c*x**3 - 6720*a**3*b**3*d*x**4 + 9240*a**2*b**4*c*x**4 - 5040*a**2*b**4*d*x**5 + 6930*a*b**5*c*x**5)/(384* sqrt(a + b*x)*a**7*x**4*(a + b*x))