Integrand size = 22, antiderivative size = 229 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx=-\frac {a \sqrt {a+b x^2}}{4 c x^4}+\frac {a d \sqrt {a+b x^2}}{3 c^2 x^3}-\frac {\left (5 b c^2+4 a d^2\right ) \sqrt {a+b x^2}}{8 c^3 x^2}+\frac {d \left (4 b c^2+3 a d^2\right ) \sqrt {a+b x^2}}{3 c^4 x}+\frac {d \left (b c^2+a d^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^5}-\frac {\left (3 b^2 c^4+12 a b c^2 d^2+8 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a} c^5} \] Output:
-1/4*a*(b*x^2+a)^(1/2)/c/x^4+1/3*a*d*(b*x^2+a)^(1/2)/c^2/x^3-1/8*(4*a*d^2+ 5*b*c^2)*(b*x^2+a)^(1/2)/c^3/x^2+1/3*d*(3*a*d^2+4*b*c^2)*(b*x^2+a)^(1/2)/c ^4/x+d*(a*d^2+b*c^2)^(3/2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2 +a)^(1/2))/c^5-1/8*(8*a^2*d^4+12*a*b*c^2*d^2+3*b^2*c^4)*arctanh((b*x^2+a)^ (1/2)/a^(1/2))/a^(1/2)/c^5
Time = 1.35 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx=\frac {\frac {c \sqrt {a+b x^2} \left (b c^2 x^2 (-15 c+32 d x)+a \left (-6 c^3+8 c^2 d x-12 c d^2 x^2+24 d^3 x^3\right )\right )}{x^4}+48 d \left (-b c^2-a d^2\right )^{3/2} \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+\frac {6 \left (3 b^2 c^4+12 a b c^2 d^2+8 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{24 c^5} \] Input:
Integrate[(a + b*x^2)^(3/2)/(x^5*(c + d*x)),x]
Output:
((c*Sqrt[a + b*x^2]*(b*c^2*x^2*(-15*c + 32*d*x) + a*(-6*c^3 + 8*c^2*d*x - 12*c*d^2*x^2 + 24*d^3*x^3)))/x^4 + 48*d*(-(b*c^2) - a*d^2)^(3/2)*ArcTan[(S qrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] + (6*(3*b^2* c^4 + 12*a*b*c^2*d^2 + 8*a^2*d^4)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sq rt[a]])/Sqrt[a])/(24*c^5)
Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(229)=458\).
Time = 1.26 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 617 |
| \(\displaystyle \int \left (-\frac {d^5 \left (a+b x^2\right )^{3/2}}{c^5 (c+d x)}+\frac {d^4 \left (a+b x^2\right )^{3/2}}{c^5 x}-\frac {d^3 \left (a+b x^2\right )^{3/2}}{c^4 x^2}+\frac {d^2 \left (a+b x^2\right )^{3/2}}{c^3 x^3}-\frac {d \left (a+b x^2\right )^{3/2}}{c^2 x^4}+\frac {\left (a+b x^2\right )^{3/2}}{c x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{3/2} d^4 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^5}-\frac {b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{c^2}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a} c}-\frac {3 a \sqrt {b} d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 c^4}-\frac {3 \sqrt {a} b d^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 c^3}+\frac {d \left (a d^2+b c^2\right )^{3/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^5}+\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a d^2+2 b c^2\right )}{2 c^4}+\frac {a d^4 \sqrt {a+b x^2}}{c^5}+\frac {d^3 \left (a+b x^2\right )^{3/2}}{c^4 x}-\frac {3 b d^3 x \sqrt {a+b x^2}}{2 c^4}-\frac {d^2 \left (a+b x^2\right )^{3/2}}{2 c^3 x^2}+\frac {3 b d^2 \sqrt {a+b x^2}}{2 c^3}+\frac {b d \sqrt {a+b x^2}}{c^2 x}+\frac {d \left (a+b x^2\right )^{3/2}}{3 c^2 x^3}-\frac {d^2 \sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )-b c d x\right )}{2 c^5}-\frac {3 b \sqrt {a+b x^2}}{8 c x^2}-\frac {\left (a+b x^2\right )^{3/2}}{4 c x^4}\) |
Input:
Int[(a + b*x^2)^(3/2)/(x^5*(c + d*x)),x]
Output:
(3*b*d^2*Sqrt[a + b*x^2])/(2*c^3) + (a*d^4*Sqrt[a + b*x^2])/c^5 - (3*b*Sqr t[a + b*x^2])/(8*c*x^2) + (b*d*Sqrt[a + b*x^2])/(c^2*x) - (3*b*d^3*x*Sqrt[ a + b*x^2])/(2*c^4) - (d^2*(2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a + b*x^2])/ (2*c^5) - (a + b*x^2)^(3/2)/(4*c*x^4) + (d*(a + b*x^2)^(3/2))/(3*c^2*x^3) - (d^2*(a + b*x^2)^(3/2))/(2*c^3*x^2) + (d^3*(a + b*x^2)^(3/2))/(c^4*x) - (b^(3/2)*d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/c^2 - (3*a*Sqrt[b]*d^3*Ar cTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^4) + (Sqrt[b]*d*(2*b*c^2 + 3*a*d^ 2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^4) + (d*(b*c^2 + a*d^2)^(3/2 )*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/c^5 - (3*b ^2*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*Sqrt[a]*c) - (3*Sqrt[a]*b*d^2*ArcT anh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*c^3) - (a^(3/2)*d^4*ArcTanh[Sqrt[a + b*x^ 2]/Sqrt[a]])/c^5
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
Time = 0.45 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-24 a \,x^{3} d^{3}-32 b \,c^{2} d \,x^{3}+12 a \,d^{2} x^{2} c +15 b \,c^{3} x^{2}-8 a d x \,c^{2}+6 c^{3} a \right )}{24 c^{4} x^{4}}-\frac {\frac {\left (8 a^{2} d^{4}+12 b \,c^{2} d^{2} a +3 b^{2} c^{4}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}-\frac {8 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{8 c^{4}}\) | \(286\) |
| default | \(\frac {-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}}{c}+\frac {d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{c^{3}}+\frac {d^{4} \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{c^{5}}-\frac {d \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )}{c^{2}}-\frac {d^{3} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{c^{4}}-\frac {d^{4} \left (\frac {\left (b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}\right )^{\frac {3}{2}}}{3}-\frac {b c \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{8 b^{\frac {3}{2}}}\right )}{d}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{2}}\right )}{c^{5}}\) | \(935\) |
Input:
int((b*x^2+a)^(3/2)/x^5/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-1/24*(b*x^2+a)^(1/2)*(-24*a*d^3*x^3-32*b*c^2*d*x^3+12*a*c*d^2*x^2+15*b*c^ 3*x^2-8*a*c^2*d*x+6*a*c^3)/c^4/x^4-1/8/c^4*((8*a^2*d^4+12*a*b*c^2*d^2+3*b^ 2*c^4)/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-8*(a^2*d^4+2*a*b*c^ 2*d^2+b^2*c^4)/c/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d *(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b *c^2)/d^2)^(1/2))/(x+c/d)))
Time = 0.28 (sec) , antiderivative size = 1061, normalized size of antiderivative = 4.63 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)/x^5/(d*x+c),x, algorithm="fricas")
Output:
[1/48*(24*(a*b*c^2*d + a^2*d^3)*sqrt(b*c^2 + a*d^2)*x^4*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a*d^2)*( b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 3*(3*b^2*c^4 + 12*a*b*c^2*d^2 + 8*a^2*d^4)*sqrt(a)*x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sq rt(a) + 2*a)/x^2) + 2*(8*a^2*c^3*d*x - 6*a^2*c^4 + 8*(4*a*b*c^3*d + 3*a^2* c*d^3)*x^3 - 3*(5*a*b*c^4 + 4*a^2*c^2*d^2)*x^2)*sqrt(b*x^2 + a))/(a*c^5*x^ 4), 1/48*(48*(a*b*c^2*d + a^2*d^3)*sqrt(-b*c^2 - a*d^2)*x^4*arctan(sqrt(-b *c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + 3*(3*b^2*c^4 + 12*a*b*c^2*d^2 + 8*a^2*d^4)*sqrt(a)*x^4* log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(8*a^2*c^3*d*x - 6 *a^2*c^4 + 8*(4*a*b*c^3*d + 3*a^2*c*d^3)*x^3 - 3*(5*a*b*c^4 + 4*a^2*c^2*d^ 2)*x^2)*sqrt(b*x^2 + a))/(a*c^5*x^4), 1/24*(3*(3*b^2*c^4 + 12*a*b*c^2*d^2 + 8*a^2*d^4)*sqrt(-a)*x^4*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 12*(a*b*c^2 *d + a^2*d^3)*sqrt(b*c^2 + a*d^2)*x^4*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d ^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt( b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + (8*a^2*c^3*d*x - 6*a^2*c^4 + 8*(4 *a*b*c^3*d + 3*a^2*c*d^3)*x^3 - 3*(5*a*b*c^4 + 4*a^2*c^2*d^2)*x^2)*sqrt(b* x^2 + a))/(a*c^5*x^4), 1/24*(24*(a*b*c^2*d + a^2*d^3)*sqrt(-b*c^2 - a*d^2) *x^4*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + 3*(3*b^2*c^4 + 12*a*b*c^2*d^2 + 8...
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{5} \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a)**(3/2)/x**5/(d*x+c),x)
Output:
Integral((a + b*x**2)**(3/2)/(x**5*(c + d*x)), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )} x^{5}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)/x^5/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)/((d*x + c)*x^5), x)
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (200) = 400\).
Time = 0.15 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.67 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx=-\frac {2 \, {\left (b^{2} c^{4} d + 2 \, a b c^{2} d^{3} + a^{2} d^{5}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} c^{5}} + \frac {{\left (3 \, b^{2} c^{4} + 12 \, a b c^{2} d^{2} + 8 \, a^{2} d^{4}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} c^{5}} + \frac {15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} b^{2} c^{3} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a b c d^{2} - 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {3}{2}} c^{2} d - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{3} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a b^{2} c^{3} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{2} b c d^{2} + 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c^{2} d + 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{3} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{2} b^{2} c^{3} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{3} b c d^{2} - 80 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c^{2} d - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{3} + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{3} b^{2} c^{3} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{4} b c d^{2} + 32 \, a^{4} b^{\frac {3}{2}} c^{2} d + 24 \, a^{5} \sqrt {b} d^{3}}{12 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{4} c^{4}} \] Input:
integrate((b*x^2+a)^(3/2)/x^5/(d*x+c),x, algorithm="giac")
Output:
-2*(b^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d^5)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c^2 - a*d^2)*c^5) + 1/ 4*(3*b^2*c^4 + 12*a*b*c^2*d^2 + 8*a^2*d^4)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*c^5) + 1/12*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^7 *b^2*c^3 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a*b*c*d^2 - 48*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c^2*d - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^ 2*sqrt(b)*d^3 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a*b^2*c^3 - 12*(sqrt(b)* x - sqrt(b*x^2 + a))^5*a^2*b*c*d^2 + 96*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^ 2*b^(3/2)*c^2*d + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b)*d^3 + 9*( sqrt(b)*x - sqrt(b*x^2 + a))^3*a^2*b^2*c^3 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^3*b*c*d^2 - 80*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(3/2)*c^2*d - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*d^3 + 15*(sqrt(b)*x - sqrt (b*x^2 + a))*a^3*b^2*c^3 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))*a^4*b*c*d^2 + 32*a^4*b^(3/2)*c^2*d + 24*a^5*sqrt(b)*d^3)/(((sqrt(b)*x - sqrt(b*x^2 + a)) ^2 - a)^4*c^4)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^5\,\left (c+d\,x\right )} \,d x \] Input:
int((a + b*x^2)^(3/2)/(x^5*(c + d*x)),x)
Output:
int((a + b*x^2)^(3/2)/(x^5*(c + d*x)), x)
Time = 0.24 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^5 (c+d x)} \, dx=\frac {48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a^{2} d^{3} x^{4}+48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (-\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b \,c^{2} d \,x^{4}-48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a^{2} d^{3} x^{4}-48 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b \,c^{2} d \,x^{4}-12 \sqrt {b \,x^{2}+a}\, a^{2} c^{4}+16 \sqrt {b \,x^{2}+a}\, a^{2} c^{3} d x -24 \sqrt {b \,x^{2}+a}\, a^{2} c^{2} d^{2} x^{2}+48 \sqrt {b \,x^{2}+a}\, a^{2} c \,d^{3} x^{3}-30 \sqrt {b \,x^{2}+a}\, a b \,c^{4} x^{2}+64 \sqrt {b \,x^{2}+a}\, a b \,c^{3} d \,x^{3}+24 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) a^{2} d^{4} x^{4}+36 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) a b \,c^{2} d^{2} x^{4}+9 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}-\sqrt {a}\right ) b^{2} c^{4} x^{4}-24 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) a^{2} d^{4} x^{4}-36 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) a b \,c^{2} d^{2} x^{4}-9 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}+\sqrt {a}\right ) b^{2} c^{4} x^{4}}{48 a \,c^{5} x^{4}} \] Input:
int((b*x^2+a)^(3/2)/x^5/(d*x+c),x)
Output:
(48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*d**3*x**4 + 48*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x **2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**2*d*x**4 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*d**3*x**4 - 48*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d*x**4 - 12*sqrt(a + b*x**2)*a**2*c**4 + 16*sqrt(a + b*x**2) *a**2*c**3*d*x - 24*sqrt(a + b*x**2)*a**2*c**2*d**2*x**2 + 48*sqrt(a + b*x **2)*a**2*c*d**3*x**3 - 30*sqrt(a + b*x**2)*a*b*c**4*x**2 + 64*sqrt(a + b* x**2)*a*b*c**3*d*x**3 + 24*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a**2*d* *4*x**4 + 36*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*a*b*c**2*d**2*x**4 + 9*sqrt(a)*log(sqrt(a + b*x**2) - sqrt(a))*b**2*c**4*x**4 - 24*sqrt(a)*log( sqrt(a + b*x**2) + sqrt(a))*a**2*d**4*x**4 - 36*sqrt(a)*log(sqrt(a + b*x** 2) + sqrt(a))*a*b*c**2*d**2*x**4 - 9*sqrt(a)*log(sqrt(a + b*x**2) + sqrt(a ))*b**2*c**4*x**4)/(48*a*c**5*x**4)