Integrand size = 24, antiderivative size = 291 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx=-\frac {\left (64 b^3 c^3-144 a b^2 c^2 d+88 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a x+b x^2}}{64 b d^4}+\frac {(4 b c-5 a d) (4 b c-a d) x \sqrt {a x+b x^2}}{32 d^3}-\frac {(8 b c-5 a d) \left (a x+b x^2\right )^{3/2}}{24 d^2}+\frac {\left (a x+b x^2\right )^{5/2}}{4 d x}+\frac {\left (128 b^4 c^4-320 a b^3 c^3 d+240 a^2 b^2 c^2 d^2-40 a^3 b c d^3-5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{64 b^{3/2} d^5}-\frac {2 c^{3/2} (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^5} \] Output:
-1/64*(-5*a^3*d^3+88*a^2*b*c*d^2-144*a*b^2*c^2*d+64*b^3*c^3)*(b*x^2+a*x)^( 1/2)/b/d^4+1/32*(-5*a*d+4*b*c)*(-a*d+4*b*c)*x*(b*x^2+a*x)^(1/2)/d^3-1/24*( -5*a*d+8*b*c)*(b*x^2+a*x)^(3/2)/d^2+1/4*(b*x^2+a*x)^(5/2)/d/x+1/64*(-5*a^4 *d^4-40*a^3*b*c*d^3+240*a^2*b^2*c^2*d^2-320*a*b^3*c^3*d+128*b^4*c^4)*arcta nh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(3/2)/d^5-2*c^(3/2)*(-a*d+b*c)^(5/2)*arc tanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/d^5
Result contains complex when optimal does not.
Time = 3.80 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\frac {(x (a+b x))^{5/2} \left (\sqrt {b} d \sqrt {x} \sqrt {a+b x} \left (15 a^3 d^3+2 a^2 b d^2 (-132 c+59 d x)+8 a b^2 d \left (54 c^2-26 c d x+17 d^2 x^2\right )-16 b^3 \left (12 c^3-6 c^2 d x+4 c d^2 x^2-3 d^3 x^3\right )\right )+384 \sqrt {b} \sqrt {c} (b c-a d)^2 \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+384 \sqrt {b} \sqrt {c} (b c-a d)^2 \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+6 \left (128 b^4 c^4-320 a b^3 c^3 d+240 a^2 b^2 c^2 d^2-40 a^3 b c d^3-5 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{192 b^{3/2} d^5 x^{5/2} (a+b x)^{5/2}} \] Input:
Integrate[(a*x + b*x^2)^(5/2)/(x*(c + d*x)),x]
Output:
((x*(a + b*x))^(5/2)*(Sqrt[b]*d*Sqrt[x]*Sqrt[a + b*x]*(15*a^3*d^3 + 2*a^2* b*d^2*(-132*c + 59*d*x) + 8*a*b^2*d*(54*c^2 - 26*c*d*x + 17*d^2*x^2) - 16* b^3*(12*c^3 - 6*c^2*d*x + 4*c*d^2*x^2 - 3*d^3*x^3)) + 384*Sqrt[b]*Sqrt[c]* (b*c - a*d)^2*(b*c - a*d - I*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[-(b*c) + 2*a *d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(-Sqrt[a] + Sqrt[a + b*x]))] + 384*Sqrt[b]*Sqrt[c]*(b*c - a*d)^2*(b*c - a*d + I*Sqrt[a ]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqr t[b*c - a*d]]*ArcTan[(Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(-Sqrt[a] + Sqrt[a + b*x]))] + 6*(128*b^4*c^4 - 320*a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTa nh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(192*b^(3/2)*d^5*x^(5/2 )*(a + b*x)^(5/2))
Time = 0.97 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.22, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {1261, 112, 27, 171, 27, 171, 27, 171, 27, 175, 65, 104, 219, 221}
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.
| \(\displaystyle \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \int \frac {x^{3/2} (a+b x)^{5/2}}{c+d x}dx}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\int \frac {\sqrt {x} (a+b x)^{3/2} (3 a c+(8 b c-5 a d) x)}{2 (c+d x)}dx}{4 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\int \frac {\sqrt {x} (a+b x)^{3/2} (3 a c+(8 b c-5 a d) x)}{c+d x}dx}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\int -\frac {(a+b x)^{3/2} \left (a c (8 b c-5 a d)+\left (48 b^2 c^2-40 a b d c-5 a^2 d^2\right ) x\right )}{2 \sqrt {x} (c+d x)}dx}{3 b d}+\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\int \frac {(a+b x)^{3/2} \left (a c (8 b c-5 a d)+\left (48 b^2 c^2-40 a b d c-5 a^2 d^2\right ) x\right )}{\sqrt {x} (c+d x)}dx}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\int -\frac {3 \sqrt {a+b x} \left (a c (4 b c-5 a d) (4 b c-a d)+\left (64 b^3 c^3-112 a b^2 d c^2+40 a^2 b d^2 c+5 a^3 d^3\right ) x\right )}{2 \sqrt {x} (c+d x)}dx}{2 d}+\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \int \frac {\sqrt {a+b x} \left (a c (4 b c-5 a d) (4 b c-a d)+\left (64 b^3 c^3-112 a b^2 d c^2+40 a^2 b d^2 c+5 a^3 d^3\right ) x\right )}{\sqrt {x} (c+d x)}dx}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \left (\frac {\int -\frac {a c \left (64 b^3 c^3-144 a b^2 d c^2+88 a^2 b d^2 c-5 a^3 d^3\right )+\left (128 b^4 c^4-320 a b^3 d c^3+240 a^2 b^2 d^2 c^2-40 a^3 b d^3 c-5 a^4 d^4\right ) x}{2 \sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}+\frac {\sqrt {x} \sqrt {a+b x} \left (5 a^3 d^3+40 a^2 b c d^2-112 a b^2 c^2 d+64 b^3 c^3\right )}{d}\right )}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} \left (5 a^3 d^3+40 a^2 b c d^2-112 a b^2 c^2 d+64 b^3 c^3\right )}{d}-\frac {\int \frac {a c \left (64 b^3 c^3-144 a b^2 d c^2+88 a^2 b d^2 c-5 a^3 d^3\right )+\left (128 b^4 c^4-320 a b^3 d c^3+240 a^2 b^2 d^2 c^2-40 a^3 b d^3 c-5 a^4 d^4\right ) x}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{2 d}\right )}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} \left (5 a^3 d^3+40 a^2 b c d^2-112 a b^2 c^2 d+64 b^3 c^3\right )}{d}-\frac {\frac {\left (-5 a^4 d^4-40 a^3 b c d^3+240 a^2 b^2 c^2 d^2-320 a b^3 c^3 d+128 b^4 c^4\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {128 b c^2 (b c-a d)^3 \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}}{2 d}\right )}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} \left (5 a^3 d^3+40 a^2 b c d^2-112 a b^2 c^2 d+64 b^3 c^3\right )}{d}-\frac {\frac {2 \left (-5 a^4 d^4-40 a^3 b c d^3+240 a^2 b^2 c^2 d^2-320 a b^3 c^3 d+128 b^4 c^4\right ) \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}-\frac {128 b c^2 (b c-a d)^3 \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}}{2 d}\right )}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} \left (5 a^3 d^3+40 a^2 b c d^2-112 a b^2 c^2 d+64 b^3 c^3\right )}{d}-\frac {\frac {2 \left (-5 a^4 d^4-40 a^3 b c d^3+240 a^2 b^2 c^2 d^2-320 a b^3 c^3 d+128 b^4 c^4\right ) \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}-\frac {256 b c^2 (b c-a d)^3 \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{2 d}\right )}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} \left (5 a^3 d^3+40 a^2 b c d^2-112 a b^2 c^2 d+64 b^3 c^3\right )}{d}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \left (-5 a^4 d^4-40 a^3 b c d^3+240 a^2 b^2 c^2 d^2-320 a b^3 c^3 d+128 b^4 c^4\right )}{\sqrt {b} d}-\frac {256 b c^2 (b c-a d)^3 \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{2 d}\right )}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {x^{3/2} (a+b x)^{5/2}}{4 d}-\frac {\frac {\sqrt {x} (a+b x)^{5/2} (8 b c-5 a d)}{3 b d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} \left (-5 a^2 d^2-40 a b c d+48 b^2 c^2\right )}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} \left (5 a^3 d^3+40 a^2 b c d^2-112 a b^2 c^2 d+64 b^3 c^3\right )}{d}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \left (-5 a^4 d^4-40 a^3 b c d^3+240 a^2 b^2 c^2 d^2-320 a b^3 c^3 d+128 b^4 c^4\right )}{\sqrt {b} d}-\frac {256 b c^{3/2} (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{d}}{2 d}\right )}{4 d}}{6 b d}}{8 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
Input:
Int[(a*x + b*x^2)^(5/2)/(x*(c + d*x)),x]
Output:
((a*x + b*x^2)^(5/2)*((x^(3/2)*(a + b*x)^(5/2))/(4*d) - (((8*b*c - 5*a*d)* Sqrt[x]*(a + b*x)^(5/2))/(3*b*d) - (((48*b^2*c^2 - 40*a*b*c*d - 5*a^2*d^2) *Sqrt[x]*(a + b*x)^(3/2))/(2*d) - (3*(((64*b^3*c^3 - 112*a*b^2*c^2*d + 40* a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[x]*Sqrt[a + b*x])/d - ((2*(128*b^4*c^4 - 320 *a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[( Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(Sqrt[b]*d) - (256*b*c^(3/2)*(b*c - a*d)^ (5/2)*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/d)/(2*d) ))/(4*d))/(6*b*d))/(8*d)))/(x^(5/2)*(a + b*x)^(5/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 0.64 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (\frac {128 c^{2} \left (a^{3} b^{\frac {3}{2}} d^{3}-3 a^{2} b^{\frac {5}{2}} c \,d^{2}+3 a \,b^{\frac {7}{2}} c^{2} d -b^{\frac {9}{2}} c^{3}\right ) \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{5}+\left (\left (a^{4} d^{4}+8 a^{3} b c \,d^{3}-48 a^{2} b^{2} c^{2} d^{2}+64 a \,b^{3} c^{3} d -\frac {128}{5} c^{4} b^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-d \left (\frac {16 \left (d^{3} x^{3}-\frac {4}{3} c \,d^{2} x^{2}+2 c^{2} d x -4 c^{3}\right ) b^{\frac {7}{2}}}{5}+\left (\frac {8 \left (\frac {17}{3} d^{2} x^{2}-\frac {26}{3} c d x +18 c^{2}\right ) b^{\frac {5}{2}}}{5}+d a \left (\frac {2 \left (\frac {59 d x}{3}-44 c \right ) b^{\frac {3}{2}}}{5}+\sqrt {b}\, a d \right )\right ) d a \right ) \sqrt {x \left (b x +a \right )}\right ) \sqrt {c \left (a d -b c \right )}\right )}{64 b^{\frac {3}{2}} \sqrt {c \left (a d -b c \right )}\, d^{5}}\) | \(275\) |
| risch | \(\frac {\left (48 b^{3} d^{3} x^{3}+136 a \,b^{2} d^{3} x^{2}-64 b^{3} c \,d^{2} x^{2}+118 a^{2} b \,d^{3} x -208 a \,b^{2} c \,d^{2} x +96 b^{3} c^{2} d x +15 a^{3} d^{3}-264 a^{2} b c \,d^{2}+432 a \,b^{2} c^{2} d -192 b^{3} c^{3}\right ) x \left (b x +a \right )}{192 b \,d^{4} \sqrt {x \left (b x +a \right )}}-\frac {\frac {\left (5 a^{4} d^{4}+40 a^{3} b c \,d^{3}-240 a^{2} b^{2} c^{2} d^{2}+320 a \,b^{3} c^{3} d -128 c^{4} b^{4}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {128 c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{128 d^{4} b}\) | \(390\) |
| default | \(\text {Expression too large to display}\) | \(1035\) |
Input:
int((b*x^2+a*x)^(5/2)/x/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-5/64/b^(3/2)/(c*(a*d-b*c))^(1/2)*(128/5*c^2*(a^3*b^(3/2)*d^3-3*a^2*b^(5/2 )*c*d^2+3*a*b^(7/2)*c^2*d-b^(9/2)*c^3)*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a* d-b*c))^(1/2))+((a^4*d^4+8*a^3*b*c*d^3-48*a^2*b^2*c^2*d^2+64*a*b^3*c^3*d-1 28/5*c^4*b^4)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-d*(16/5*(d^3*x^3-4/3*c* d^2*x^2+2*c^2*d*x-4*c^3)*b^(7/2)+(8/5*(17/3*d^2*x^2-26/3*c*d*x+18*c^2)*b^( 5/2)+d*a*(2/5*(59/3*d*x-44*c)*b^(3/2)+b^(1/2)*a*d))*d*a)*(x*(b*x+a))^(1/2) )*(c*(a*d-b*c))^(1/2))/d^5
Time = 0.70 (sec) , antiderivative size = 1216, normalized size of antiderivative = 4.18 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a*x)^(5/2)/x/(d*x+c),x, algorithm="fricas")
Output:
[-1/384*(3*(128*b^4*c^4 - 320*a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 40*a^3*b *c*d^3 - 5*a^4*d^4)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 384*(b^4*c^3 - 2*a*b^3*c^2*d + a^2*b^2*c*d^2)*sqrt(b*c^2 - a*c*d)*log((a* c + (2*b*c - a*d)*x - 2*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) - 2*(48*b^4*d^4*x^3 - 192*b^4*c^3*d + 432*a*b^3*c^2*d^2 - 264*a^2*b^2*c*d^ 3 + 15*a^3*b*d^4 - 8*(8*b^4*c*d^3 - 17*a*b^3*d^4)*x^2 + 2*(48*b^4*c^2*d^2 - 104*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x)*sqrt(b*x^2 + a*x))/(b^2*d^5), 1/384 *(768*(b^4*c^3 - 2*a*b^3*c^2*d + a^2*b^2*c*d^2)*sqrt(-b*c^2 + a*c*d)*arcta n(sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a*x)/(b*c*x + a*c)) - 3*(128*b^4*c^4 - 320*a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt( b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(48*b^4*d^4*x^3 - 192* b^4*c^3*d + 432*a*b^3*c^2*d^2 - 264*a^2*b^2*c*d^3 + 15*a^3*b*d^4 - 8*(8*b^ 4*c*d^3 - 17*a*b^3*d^4)*x^2 + 2*(48*b^4*c^2*d^2 - 104*a*b^3*c*d^3 + 59*a^2 *b^2*d^4)*x)*sqrt(b*x^2 + a*x))/(b^2*d^5), -1/192*(3*(128*b^4*c^4 - 320*a* b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 - 5*a^4*d^4)*sqrt(-b)*arc tan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - 192*(b^4*c^3 - 2*a*b^3*c^2*d + a^2*b^2*c*d^2)*sqrt(b*c^2 - a*c*d)*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b* c^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) - (48*b^4*d^4*x^3 - 192*b^4*c^3 *d + 432*a*b^3*c^2*d^2 - 264*a^2*b^2*c*d^3 + 15*a^3*b*d^4 - 8*(8*b^4*c*d^3 - 17*a*b^3*d^4)*x^2 + 2*(48*b^4*c^2*d^2 - 104*a*b^3*c*d^3 + 59*a^2*b^2...
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a*x)**(5/2)/x/(d*x+c),x)
Output:
Integral((x*(a + b*x))**(5/2)/(x*(c + d*x)), x)
Exception generated. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a*x)^(5/2)/x/(d*x+c),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x^2+a*x)^(5/2)/x/(d*x+c),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{5/2}}{x\,\left (c+d\,x\right )} \,d x \] Input:
int((a*x + b*x^2)^(5/2)/(x*(c + d*x)),x)
Output:
int((a*x + b*x^2)^(5/2)/(x*(c + d*x)), x)
Time = 10.78 (sec) , antiderivative size = 699, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x (c+d x)} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a*x)^(5/2)/x/(d*x+c),x)
Output:
( - 384*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b *x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*b**2*c*d**2 + 768*s qrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqr t(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b**3*c**2*d - 384*sqrt(c)*sqrt( a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d) *sqrt(b))/(sqrt(c)*sqrt(b)))*b**4*c**3 - 384*sqrt(c)*sqrt(a*d - b*c)*atan( (sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt( c)*sqrt(b)))*a**2*b**2*c*d**2 + 768*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b )))*a*b**3*c**2*d - 384*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sq rt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b**4*c** 3 + 15*sqrt(x)*sqrt(a + b*x)*a**3*b*d**4 - 264*sqrt(x)*sqrt(a + b*x)*a**2* b**2*c*d**3 + 118*sqrt(x)*sqrt(a + b*x)*a**2*b**2*d**4*x + 432*sqrt(x)*sqr t(a + b*x)*a*b**3*c**2*d**2 - 208*sqrt(x)*sqrt(a + b*x)*a*b**3*c*d**3*x + 136*sqrt(x)*sqrt(a + b*x)*a*b**3*d**4*x**2 - 192*sqrt(x)*sqrt(a + b*x)*b** 4*c**3*d + 96*sqrt(x)*sqrt(a + b*x)*b**4*c**2*d**2*x - 64*sqrt(x)*sqrt(a + b*x)*b**4*c*d**3*x**2 + 48*sqrt(x)*sqrt(a + b*x)*b**4*d**4*x**3 - 15*sqrt (b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**4*d**4 - 120*sqrt(b) *log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**3*b*c*d**3 + 720*sqrt(b )*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*b**2*c**2*d**2 - ...