Integrand size = 26, antiderivative size = 421 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {\left (d \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 d^2 e+40 b^2 c d e^2+5 b^3 e^3\right )\right )+e \left (3 A b^2 e^3 (2 c d-b e)+B d \left (96 c^3 d^3-192 b c^2 d^2 e+98 b^2 c d e^2-5 b^3 e^3\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 d^2 e^4 (c d-b e)^2 (d+e x)^2}+\frac {\left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e (B d (14 c d-11 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{24 d e^2 (c d-b e) (d+e x)^4}+\frac {2 B c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^5}+\frac {\left (3 A b^4 e^5-B d \left (128 c^4 d^4-320 b c^3 d^3 e+240 b^2 c^2 d^2 e^2-40 b^3 c d e^3-5 b^4 e^4\right )\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{128 d^{5/2} e^5 (c d-b e)^{5/2}} \]
1/24*(d*(3*A*b*e^2-B*d*(-5*b*e+8*c*d))-e*(B*d*(-11*b*e+14*c*d)-3*A*e*(-b*e +2*c*d))*x)*(c*x^2+b*x)^(3/2)/d/e^2/(-b*e+c*d)/(e*x+d)^4+2*B*c^(3/2)*arcta nh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/e^5+1/128*(3*A*b^4*e^5-B*d*(-5*b^4*e^4-40* b^3*c*d*e^3+240*b^2*c^2*d^2*e^2-320*b*c^3*d^3*e+128*c^4*d^4))*arctanh(1/2* (b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/d^(5/2)/e ^5/(-b*e+c*d)^(5/2)-1/64*(d*(3*A*b^3*e^4+B*d*(5*b^3*e^3+40*b^2*c*d*e^2-112 *b*c^2*d^2*e+64*c^3*d^3))+e*(3*A*b^2*e^3*(-b*e+2*c*d)+B*d*(-5*b^3*e^3+98*b ^2*c*d*e^2-192*b*c^2*d^2*e+96*c^3*d^3))*x)*(c*x^2+b*x)^(1/2)/d^2/e^4/(-b*e +c*d)^2/(e*x+d)^2
Time = 14.68 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {(x (b+c x))^{3/2} \left (\frac {6 (-B d+A e) x^{5/2} (b+c x)}{(d+e x)^4}+\frac {(B d (2 c d-5 b e)-3 A e (-2 c d+b e)) x^{5/2} (b+c x)}{d (c d-b e) (d+e x)^3}+\frac {\frac {\left (3 A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )+B \left (-8 c^2 d^3+5 b^2 d e^2\right )\right ) x^{5/2} (b+c x)}{4 (d+e x)^2}+\frac {-\frac {e \sqrt {x} (b+c x) \left (3 A e^4 \left (16 c^4 d^2 x^3+8 b c^3 d x^2 (3 d-2 e x)+b^4 e (3 d+2 e x)+2 b^2 c^2 x \left (d^2-13 d e x+e^2 x^2\right )+b^3 c \left (-3 d^2-4 d e x+4 e^2 x^2\right )\right )+B d \left (5 b^4 e^4 (3 d+2 e x)+8 b c^3 d^2 e \left (66 d^2+34 d e x-11 e^2 x^2\right )+5 b^3 c e^3 \left (21 d^2+12 d e x+4 e^2 x^2\right )-16 c^4 d^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+2 b^2 c^2 e^2 \left (-228 d^3-123 d^2 e x+15 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{d+e x}+\frac {384 B c^{3/2} d^3 (-c d+b e)^3 (b+c x) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {1+\frac {c x}{b}}}+3 \sqrt {d} \sqrt {c d-b e} \left (-3 A b^4 e^5+B d \left (128 c^4 d^4-320 b c^3 d^3 e+240 b^2 c^2 d^2 e^2-40 b^3 c d e^3-5 b^4 e^4\right )\right ) \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{8 e^5 (b+c x)^2}}{d^2 (c d-b e)^2}\right )}{24 d (-c d+b e) x^{3/2}} \]
((x*(b + c*x))^(3/2)*((6*(-(B*d) + A*e)*x^(5/2)*(b + c*x))/(d + e*x)^4 + ( (B*d*(2*c*d - 5*b*e) - 3*A*e*(-2*c*d + b*e))*x^(5/2)*(b + c*x))/(d*(c*d - b*e)*(d + e*x)^3) + (((3*A*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) + B*(-8*c^2 *d^3 + 5*b^2*d*e^2))*x^(5/2)*(b + c*x))/(4*(d + e*x)^2) + (-((e*Sqrt[x]*(b + c*x)*(3*A*e^4*(16*c^4*d^2*x^3 + 8*b*c^3*d*x^2*(3*d - 2*e*x) + b^4*e*(3* d + 2*e*x) + 2*b^2*c^2*x*(d^2 - 13*d*e*x + e^2*x^2) + b^3*c*(-3*d^2 - 4*d* e*x + 4*e^2*x^2)) + B*d*(5*b^4*e^4*(3*d + 2*e*x) + 8*b*c^3*d^2*e*(66*d^2 + 34*d*e*x - 11*e^2*x^2) + 5*b^3*c*e^3*(21*d^2 + 12*d*e*x + 4*e^2*x^2) - 16 *c^4*d^2*(12*d^3 + 6*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 2*b^2*c^2*e^2*(-22 8*d^3 - 123*d^2*e*x + 15*d*e^2*x^2 + 5*e^3*x^3))))/(d + e*x)) + (384*B*c^( 3/2)*d^3*(-(c*d) + b*e)^3*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(S qrt[b]*Sqrt[1 + (c*x)/b]) + 3*Sqrt[d]*Sqrt[c*d - b*e]*(-3*A*b^4*e^5 + B*d* (128*c^4*d^4 - 320*b*c^3*d^3*e + 240*b^2*c^2*d^2*e^2 - 40*b^3*c*d*e^3 - 5* b^4*e^4))*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(8*e^5*(b + c*x)^2))/(d^2*(c*d - b*e)^2)))/(24*d*(-(c*d) + b*e)* x^(3/2))
Time = 0.82 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1229, 27, 1229, 27, 1269, 1091, 219, 1154, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\int \frac {\left (b \left (3 A b e^2-B d (8 c d-5 b e)\right )-16 B c d (c d-b e) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)^3}dx}{8 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\int \frac {\left (b \left (3 A b e^2-B d (8 c d-5 b e)\right )-16 B c d (c d-b e) x\right ) \sqrt {c x^2+b x}}{(d+e x)^3}dx}{16 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (-5 b^3 e^3+98 b^2 c d e^2-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (5 b^3 e^3+40 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {\int \frac {128 B c^2 d^2 x (c d-b e)^2+b \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 e d^2+40 b^2 c e^2 d+5 b^3 e^3\right )\right )}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 d e^2 (c d-b e)}}{16 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (-5 b^3 e^3+98 b^2 c d e^2-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (5 b^3 e^3+40 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {\int \frac {128 B c^2 d^2 x (c d-b e)^2+b \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 e d^2+40 b^2 c e^2 d+5 b^3 e^3\right )\right )}{(d+e x) \sqrt {c x^2+b x}}dx}{8 d e^2 (c d-b e)}}{16 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (-5 b^3 e^3+98 b^2 c d e^2-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (5 b^3 e^3+40 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {\frac {\left (3 A b^4 e^5-B d \left (-5 b^4 e^4-40 b^3 c d e^3+240 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+128 c^4 d^4\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {128 B c^2 d^2 (c d-b e)^2 \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}}{8 d e^2 (c d-b e)}}{16 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (-5 b^3 e^3+98 b^2 c d e^2-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (5 b^3 e^3+40 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {\frac {\left (3 A b^4 e^5-B d \left (-5 b^4 e^4-40 b^3 c d e^3+240 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+128 c^4 d^4\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {256 B c^2 d^2 (c d-b e)^2 \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}}{8 d e^2 (c d-b e)}}{16 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (-5 b^3 e^3+98 b^2 c d e^2-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (5 b^3 e^3+40 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {\frac {\left (3 A b^4 e^5-B d \left (-5 b^4 e^4-40 b^3 c d e^3+240 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+128 c^4 d^4\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {256 B c^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e)^2}{e}}{8 d e^2 (c d-b e)}}{16 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (-5 b^3 e^3+98 b^2 c d e^2-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (5 b^3 e^3+40 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {\frac {256 B c^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e)^2}{e}-\frac {2 \left (3 A b^4 e^5-B d \left (-5 b^4 e^4-40 b^3 c d e^3+240 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+128 c^4 d^4\right )\right ) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}}{8 d e^2 (c d-b e)}}{16 d e^2 (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (-5 b^3 e^3+98 b^2 c d e^2-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (5 b^3 e^3+40 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {\frac {\left (3 A b^4 e^5-B d \left (-5 b^4 e^4-40 b^3 c d e^3+240 b^2 c^2 d^2 e^2-320 b c^3 d^3 e+128 c^4 d^4\right )\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}+\frac {256 B c^{3/2} d^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (c d-b e)^2}{e}}{8 d e^2 (c d-b e)}}{16 d e^2 (c d-b e)}\) |
((d*(3*A*b*e^2 - B*d*(8*c*d - 5*b*e)) - e*(B*d*(14*c*d - 11*b*e) - 3*A*e*( 2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(24*d*e^2*(c*d - b*e)*(d + e*x)^4) - (((d*(3*A*b^3*e^4 + B*d*(64*c^3*d^3 - 112*b*c^2*d^2*e + 40*b^2*c*d*e^2 + 5*b^3*e^3)) + e*(3*A*b^2*e^3*(2*c*d - b*e) + B*d*(96*c^3*d^3 - 192*b*c^2*d ^2*e + 98*b^2*c*d*e^2 - 5*b^3*e^3))*x)*Sqrt[b*x + c*x^2])/(4*d*e^2*(c*d - b*e)*(d + e*x)^2) - ((256*B*c^(3/2)*d^2*(c*d - b*e)^2*ArcTanh[(Sqrt[c]*x)/ Sqrt[b*x + c*x^2]])/e + ((3*A*b^4*e^5 - B*d*(128*c^4*d^4 - 320*b*c^3*d^3*e + 240*b^2*c^2*d^2*e^2 - 40*b^3*c*d*e^3 - 5*b^4*e^4))*ArcTanh[(b*d + (2*c* d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(Sqrt[d]*e*Sqr t[c*d - b*e]))/(8*d*e^2*(c*d - b*e)))/(16*d*e^2*(c*d - b*e))
3.12.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.71 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\left (A \,b^{4} e^{5}+\frac {5}{3} B \,b^{4} d \,e^{4}+\frac {40}{3} B \,b^{3} c \,d^{2} e^{3}-80 B \,b^{2} c^{2} d^{3} e^{2}+\frac {320}{3} B b \,c^{3} d^{4} e -\frac {128}{3} B \,c^{4} d^{5}\right ) \left (e x +d \right )^{4} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\sqrt {d \left (b e -c d \right )}\, \left (\frac {256 B \,d^{2} \left (e x +d \right )^{4} \left (-\frac {b^{2} e^{2} c^{\frac {3}{2}}}{2}+d \left (c^{\frac {5}{2}} b e -\frac {c^{\frac {7}{2}} d}{2}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{3}+\left (\frac {64 B \,c^{3} d^{7}}{3}-\frac {112 B \,c^{2} e \left (-2 c x +b \right ) d^{6}}{3}+\frac {40 c B \left (\frac {104}{15} c^{2} x^{2}-\frac {148}{15} b c x +b^{2}\right ) e^{2} d^{5}}{3}+\frac {5 B \left (\frac {80}{3} c^{3} x^{3}-\frac {296}{3} b \,c^{2} x^{2}+\frac {86}{3} b^{2} c x +b^{3}\right ) e^{3} d^{4}}{3}+\left (\left (\frac {55 B x}{9}+A \right ) b^{3}-\frac {2 c x \left (-\frac {274 B x}{3}+A \right ) b^{2}}{3}-8 c^{2} x^{2} \left (\frac {97 B x}{9}+A \right ) b -\frac {16 A \,c^{3} x^{3}}{3}\right ) e^{4} d^{3}+\frac {11 \left (\left (\frac {73 B x}{33}+A \right ) b^{2}+4 c x \left (\frac {191 B x}{66}+A \right ) b +\frac {24 A \,c^{2} x^{2}}{11}\right ) x \,e^{5} b \,d^{2}}{3}-\frac {11 x^{2} e^{6} \left (\left (\frac {5 B x}{11}+A \right ) b +\frac {2 A c x}{11}\right ) b^{2} d}{3}-A \,x^{3} b^{3} e^{7}\right ) \sqrt {x \left (c x +b \right )}\, e \right )\right )}{64 \sqrt {d \left (b e -c d \right )}\, \left (b e -c d \right )^{2} e^{5} d^{2} \left (e x +d \right )^{4}}\) | \(447\) |
| default | \(\text {Expression too large to display}\) | \(7164\) |
-3/64*((A*b^4*e^5+5/3*B*b^4*d*e^4+40/3*B*b^3*c*d^2*e^3-80*B*b^2*c^2*d^3*e^ 2+320/3*B*b*c^3*d^4*e-128/3*B*c^4*d^5)*(e*x+d)^4*arctan((x*(c*x+b))^(1/2)/ x*d/(d*(b*e-c*d))^(1/2))+(d*(b*e-c*d))^(1/2)*(256/3*B*d^2*(e*x+d)^4*(-1/2* b^2*e^2*c^(3/2)+d*(c^(5/2)*b*e-1/2*c^(7/2)*d))*arctanh((x*(c*x+b))^(1/2)/x /c^(1/2))+(64/3*B*c^3*d^7-112/3*B*c^2*e*(-2*c*x+b)*d^6+40/3*c*B*(104/15*c^ 2*x^2-148/15*b*c*x+b^2)*e^2*d^5+5/3*B*(80/3*c^3*x^3-296/3*b*c^2*x^2+86/3*b ^2*c*x+b^3)*e^3*d^4+((55/9*B*x+A)*b^3-2/3*c*x*(-274/3*B*x+A)*b^2-8*c^2*x^2 *(97/9*B*x+A)*b-16/3*A*c^3*x^3)*e^4*d^3+11/3*((73/33*B*x+A)*b^2+4*c*x*(191 /66*B*x+A)*b+24/11*A*c^2*x^2)*x*e^5*b*d^2-11/3*x^2*e^6*((5/11*B*x+A)*b+2/1 1*A*c*x)*b^2*d-A*x^3*b^3*e^7)*(x*(c*x+b))^(1/2)*e))/(d*(b*e-c*d))^(1/2)/(b *e-c*d)^2/e^5/d^2/(e*x+d)^4
Leaf count of result is larger than twice the leaf count of optimal. 1441 vs. \(2 (393) = 786\).
Time = 14.60 (sec) , antiderivative size = 5781, normalized size of antiderivative = 13.73 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Too large to display} \]
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{5}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^5} \,d x \]