Integrand size = 24, antiderivative size = 131 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\frac {2 a^2 c \sqrt {a x^2+b x^3}}{x}+\frac {2 a c \left (a x^2+b x^3\right )^{3/2}}{3 x^3}+\frac {2 c \left (a x^2+b x^3\right )^{5/2}}{5 x^5}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{7 b x^7}-2 a^{5/2} c \text {arctanh}\left (\frac {\sqrt {a x^2+b x^3}}{\sqrt {a} x}\right ) \] Output:
2*a^2*c*(b*x^3+a*x^2)^(1/2)/x+2/3*a*c*(b*x^3+a*x^2)^(3/2)/x^3+2/5*c*(b*x^3 +a*x^2)^(5/2)/x^5+2/7*d*(b*x^3+a*x^2)^(7/2)/b/x^7-2*a^(5/2)*c*arctanh((b*x ^3+a*x^2)^(1/2)/a^(1/2)/x)
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\frac {2 x \left ((a+b x) \left (15 a^3 d+3 b^3 x^2 (7 c+5 d x)+a b^2 x (77 c+45 d x)+a^2 b (161 c+45 d x)\right )-105 a^{5/2} b c \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{105 b \sqrt {x^2 (a+b x)}} \] Input:
Integrate[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x^6,x]
Output:
(2*x*((a + b*x)*(15*a^3*d + 3*b^3*x^2*(7*c + 5*d*x) + a*b^2*x*(77*c + 45*d *x) + a^2*b*(161*c + 45*d*x)) - 105*a^(5/2)*b*c*Sqrt[a + b*x]*ArcTanh[Sqrt [a + b*x]/Sqrt[a]]))/(105*b*Sqrt[x^2*(a + b*x)])
Time = 0.57 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1945, 1927, 1927, 1927, 1914, 219}
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.
| \(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{5/2} (c+d x)}{x^6} \, dx\) |
| \(\Big \downarrow \) 1945 |
| \(\displaystyle c \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^6}dx+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{7 b x^7}\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle c \left (a \int \frac {\left (b x^3+a x^2\right )^{3/2}}{x^4}dx+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 x^5}\right )+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{7 b x^7}\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle c \left (a \left (a \int \frac {\sqrt {b x^3+a x^2}}{x^2}dx+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\right )+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 x^5}\right )+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{7 b x^7}\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle c \left (a \left (a \left (a \int \frac {1}{\sqrt {b x^3+a x^2}}dx+\frac {2 \sqrt {a x^2+b x^3}}{x}\right )+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\right )+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 x^5}\right )+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{7 b x^7}\) |
| \(\Big \downarrow \) 1914 |
| \(\displaystyle c \left (a \left (a \left (\frac {2 \sqrt {a x^2+b x^3}}{x}-2 a \int \frac {1}{1-\frac {a x^2}{b x^3+a x^2}}d\frac {x}{\sqrt {b x^3+a x^2}}\right )+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\right )+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 x^5}\right )+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{7 b x^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (a \left (a \left (\frac {2 \sqrt {a x^2+b x^3}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )\right )+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\right )+\frac {2 \left (a x^2+b x^3\right )^{5/2}}{5 x^5}\right )+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{7 b x^7}\) |
Input:
Int[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x^6,x]
Output:
(2*d*(a*x^2 + b*x^3)^(7/2))/(7*b*x^7) + c*((2*(a*x^2 + b*x^3)^(5/2))/(5*x^ 5) + a*((2*(a*x^2 + b*x^3)^(3/2))/(3*x^3) + a*((2*Sqrt[a*x^2 + b*x^3])/x - 2*Sqrt[a]*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[2/(2 - n) Subst[Int[1/(1 - a*x^2), x], x, x/Sqrt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* (n - j)*(p/(c^j*(m + n*p + 1))) Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) , x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)) Int[(e* x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p *(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
Time = 0.58 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(\frac {2 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}} \left (15 d \left (b x +a \right )^{\frac {7}{2}}-105 a^{\frac {5}{2}} b c \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+21 \left (b x +a \right )^{\frac {5}{2}} b c +35 \left (b x +a \right )^{\frac {3}{2}} a b c +105 \sqrt {b x +a}\, a^{2} b c \right )}{105 x^{5} \left (b x +a \right )^{\frac {5}{2}} b}\) | \(96\) |
| pseudoelliptic | \(\frac {\frac {5 b^{4} x^{5} \left (a d -\frac {3 b c}{10}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{64}-\frac {31 \left (\frac {5 b^{3} x^{3} \left (5 d x +c \right ) a^{\frac {3}{2}}}{124}+b^{2} x^{2} \left (\frac {295 d x}{186}+c \right ) a^{\frac {5}{2}}+\frac {42 x \left (\frac {85 d x}{63}+c \right ) b \,a^{\frac {7}{2}}}{31}+\frac {4 \left (5 d x +4 c \right ) a^{\frac {9}{2}}}{31}-\frac {15 \sqrt {a}\, b^{4} c \,x^{4}}{248}\right ) \sqrt {b x +a}}{80}}{a^{\frac {5}{2}} x^{5}}\) | \(118\) |
Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^6,x,method=_RETURNVERBOSE)
Output:
2/105*(b*x^3+a*x^2)^(5/2)*(15*d*(b*x+a)^(7/2)-105*a^(5/2)*b*c*arctanh((b*x +a)^(1/2)/a^(1/2))+21*(b*x+a)^(5/2)*b*c+35*(b*x+a)^(3/2)*a*b*c+105*(b*x+a) ^(1/2)*a^2*b*c)/x^5/(b*x+a)^(5/2)/b
Time = 0.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.92 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\left [\frac {105 \, a^{\frac {5}{2}} b c x \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (15 \, b^{3} d x^{3} + 161 \, a^{2} b c + 15 \, a^{3} d + 3 \, {\left (7 \, b^{3} c + 15 \, a b^{2} d\right )} x^{2} + {\left (77 \, a b^{2} c + 45 \, a^{2} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{105 \, b x}, \frac {2 \, {\left (105 \, \sqrt {-a} a^{2} b c x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{b x^{2} + a x}\right ) + {\left (15 \, b^{3} d x^{3} + 161 \, a^{2} b c + 15 \, a^{3} d + 3 \, {\left (7 \, b^{3} c + 15 \, a b^{2} d\right )} x^{2} + {\left (77 \, a b^{2} c + 45 \, a^{2} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}\right )}}{105 \, b x}\right ] \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^6,x, algorithm="fricas")
Output:
[1/105*(105*a^(5/2)*b*c*x*log((b*x^2 + 2*a*x - 2*sqrt(b*x^3 + a*x^2)*sqrt( a))/x^2) + 2*(15*b^3*d*x^3 + 161*a^2*b*c + 15*a^3*d + 3*(7*b^3*c + 15*a*b^ 2*d)*x^2 + (77*a*b^2*c + 45*a^2*b*d)*x)*sqrt(b*x^3 + a*x^2))/(b*x), 2/105* (105*sqrt(-a)*a^2*b*c*x*arctan(sqrt(b*x^3 + a*x^2)*sqrt(-a)/(b*x^2 + a*x)) + (15*b^3*d*x^3 + 161*a^2*b*c + 15*a^3*d + 3*(7*b^3*c + 15*a*b^2*d)*x^2 + (77*a*b^2*c + 45*a^2*b*d)*x)*sqrt(b*x^3 + a*x^2))/(b*x)]
\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}} \left (c + d x\right )}{x^{6}}\, dx \] Input:
integrate((d*x+c)*(b*x**3+a*x**2)**(5/2)/x**6,x)
Output:
Integral((x**2*(a + b*x))**(5/2)*(c + d*x)/x**6, x)
\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {5}{2}} {\left (d x + c\right )}}{x^{6}} \,d x } \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^6,x, algorithm="maxima")
Output:
integrate((b*x^3 + a*x^2)^(5/2)*(d*x + c)/x^6, x)
Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.15 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\frac {2 \, a^{3} c \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} - \frac {2 \, {\left (105 \, a^{3} b c \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 161 \, \sqrt {-a} a^{\frac {5}{2}} b c + 15 \, \sqrt {-a} a^{\frac {7}{2}} d\right )} \mathrm {sgn}\left (x\right )}{105 \, \sqrt {-a} b} + \frac {2 \, {\left (21 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{7} c \mathrm {sgn}\left (x\right ) + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{7} c \mathrm {sgn}\left (x\right ) + 105 \, \sqrt {b x + a} a^{2} b^{7} c \mathrm {sgn}\left (x\right ) + 15 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{6} d \mathrm {sgn}\left (x\right )\right )}}{105 \, b^{7}} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^6,x, algorithm="giac")
Output:
2*a^3*c*arctan(sqrt(b*x + a)/sqrt(-a))*sgn(x)/sqrt(-a) - 2/105*(105*a^3*b* c*arctan(sqrt(a)/sqrt(-a)) + 161*sqrt(-a)*a^(5/2)*b*c + 15*sqrt(-a)*a^(7/2 )*d)*sgn(x)/(sqrt(-a)*b) + 2/105*(21*(b*x + a)^(5/2)*b^7*c*sgn(x) + 35*(b* x + a)^(3/2)*a*b^7*c*sgn(x) + 105*sqrt(b*x + a)*a^2*b^7*c*sgn(x) + 15*(b*x + a)^(7/2)*b^6*d*sgn(x))/b^7
Timed out. \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{5/2}\,\left (c+d\,x\right )}{x^6} \,d x \] Input:
int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/x^6,x)
Output:
int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/x^6, x)
Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^6} \, dx=\frac {30 \sqrt {b x +a}\, a^{3} d +322 \sqrt {b x +a}\, a^{2} b c +90 \sqrt {b x +a}\, a^{2} b d x +154 \sqrt {b x +a}\, a \,b^{2} c x +90 \sqrt {b x +a}\, a \,b^{2} d \,x^{2}+42 \sqrt {b x +a}\, b^{3} c \,x^{2}+30 \sqrt {b x +a}\, b^{3} d \,x^{3}+105 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} b c -105 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} b c}{105 b} \] Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^6,x)
Output:
(30*sqrt(a + b*x)*a**3*d + 322*sqrt(a + b*x)*a**2*b*c + 90*sqrt(a + b*x)*a **2*b*d*x + 154*sqrt(a + b*x)*a*b**2*c*x + 90*sqrt(a + b*x)*a*b**2*d*x**2 + 42*sqrt(a + b*x)*b**3*c*x**2 + 30*sqrt(a + b*x)*b**3*d*x**3 + 105*sqrt(a )*log(sqrt(a + b*x) - sqrt(a))*a**2*b*c - 105*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*a**2*b*c)/(105*b)