Integrand size = 26, antiderivative size = 495 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {5 \left (A e \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )-2 B d \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )+e \left (4 A c e (2 c d-b e)-B \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{8 e^6 (d+e x)}-\frac {5 (4 B d (2 c d-b e)-A e (4 c d-b e)+e (4 B c d-b B e-2 A c e) x) \left (b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac {(2 B d-A e+B e x) \left (b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac {5 \left (2 A c e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+24 b^2 c d e^2-b^3 e^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 \sqrt {c} e^7}+\frac {5 \left (B d \left (64 c^3 d^3-112 b c^2 d^2 e+56 b^2 c d e^2-7 b^3 e^3\right )-A e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 \sqrt {d} e^7 \sqrt {c d-b e}} \]
-5/12*(4*B*d*(-b*e+2*c*d)-A*e*(-b*e+4*c*d)+e*(-2*A*c*e-B*b*e+4*B*c*d)*x)*( c*x^2+b*x)^(3/2)/e^4/(e*x+d)^2+1/3*(B*e*x-A*e+2*B*d)*(c*x^2+b*x)^(5/2)/e^2 /(e*x+d)^3+5/8*(2*A*c*e*(3*b^2*e^2-16*b*c*d*e+16*c^2*d^2)-B*(-b^3*e^3+24*b ^2*c*d*e^2-80*b*c^2*d^2*e+64*c^3*d^3))*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2) )/e^7/c^(1/2)+5/16*(B*d*(-7*b^3*e^3+56*b^2*c*d*e^2-112*b*c^2*d^2*e+64*c^3* d^3)-A*e*(-b^3*e^3+18*b^2*c*d*e^2-48*b*c^2*d^2*e+32*c^3*d^3))*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))/e^7/d^(1/ 2)/(-b*e+c*d)^(1/2)-5/8*(A*e*(b^2*e^2-12*b*c*d*e+16*c^2*d^2)-2*B*d*(3*b^2* e^2-16*b*c*d*e+16*c^2*d^2)+e*(4*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-12*b*c*d*e+1 6*c^2*d^2))*x)*(c*x^2+b*x)^(1/2)/e^6/(e*x+d)
Leaf count is larger than twice the leaf count of optimal. \(2131\) vs. \(2(495)=990\).
Time = 16.27 (sec) , antiderivative size = 2131, normalized size of antiderivative = 4.31 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Result too large to show} \]
((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(5/2))/(3*d*(-(c*d) + b*e)*(d + e*x)^3) + ((x*(b + c*x))^(5/2)*(((-4*c*d*(B*d - A*e) + (e*(7*b*B*d - 6*A*c *d - A*b*e))/2)*x^(7/2)*(b + c*x)^(7/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((((e*(24*A*c^2*d^2 + 2*b*c*d*(14*B*d - 17*A*e) - 3*b^2*e*(7*B*d - A*e)) )/4 - (5*c*d*(B*d*(8*c*d - 7*b*e) - A*e*(2*c*d - b*e)))/2)*x^(7/2)*(b + c* x)^(7/2))/(d*(-(c*d) + b*e)*(d + e*x)) + ((-3*(16*A*c^3*d^3 + 2*b^2*c*d*e* (98*B*d - 39*A*e) - 8*b*c^2*d^2*(21*B*d - 8*A*e) - 5*b^3*e^2*(7*B*d - A*e) )*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((5/(16*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (15*b^3*((2*c*x)/b - (4*c ^2*x^2)/(3*b^2) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/( Sqrt[b]*Sqrt[1 + (c*x)/b])))/(512*c^3*x^3*(1 + (c*x)/b)^3)))/(5*e) - (d*(( (5*b^3*Sqrt[x]*Sqrt[b + c*x])/(64*c) + (59*b^2*x^(3/2)*Sqrt[b + c*x])/96 + (17*b*c*x^(5/2)*Sqrt[b + c*x])/24 + (c^2*x^(7/2)*Sqrt[b + c*x])/4 - (5*b^ (7/2)*Sqrt[b + c*x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(64*c^(3/2)*Sqrt[1 + (c*x)/b]))/e - (d*(((11*b^2*Sqrt[x]*Sqrt[b + c*x])/8 + (13*b*c*x^(3/2)* Sqrt[b + c*x])/12 + (c^2*x^(5/2)*Sqrt[b + c*x])/3 + (5*b^(5/2)*Sqrt[b + c* x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(8*Sqrt[c]*Sqrt[1 + (c*x)/b]))/e - (d*((c*((5*b*Sqrt[x]*Sqrt[b + c*x])/4 + (c*x^(3/2)*Sqrt[b + c*x])/2 + (3*b ^(3/2)*Sqrt[b + c*x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(4*Sqrt[c]*Sqrt[1 + (c*x)/b])))/e - ((c*d - b*e)*((c*(Sqrt[x]*Sqrt[b + c*x] + (Sqrt[b]*S...
Time = 0.99 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1230, 27, 1230, 27, 1230, 25, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \int \frac {3 (b (2 B d-A e)+(4 B c d-b B e-2 A c e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^3}dx}{18 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \int \frac {(b (2 B d-A e)+(4 B c d-b B e-2 A c e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^3}dx}{6 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \int \frac {2 \left (b (4 B d (2 c d-b e)-A e (4 c d-b e))-\left (4 A c e (2 c d-b e)-B \left (16 c^2 d^2-12 b c e d+b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{(d+e x)^2}dx}{8 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \int \frac {\left (b (4 B d (2 c d-b e)-A e (4 c d-b e))-\left (4 A c e (2 c d-b e)-B \left (16 c^2 d^2-12 b c e d+b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{(d+e x)^2}dx}{4 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \left (-\frac {\int -\frac {b \left (-16 c^2 (2 B d-A e) d^2+4 b c e (8 B d-3 A e) d-b^2 e^2 (6 B d-A e)\right )+\left (2 A c e \left (16 c^2 d^2-16 b c e d+3 b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 e d^2+24 b^2 c e^2 d-b^3 e^3\right )\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\int \frac {b \left (-16 c^2 (2 B d-A e) d^2+4 b c e (8 B d-3 A e) d-b^2 e^2 (6 B d-A e)\right )+\left (2 A c e \left (16 c^2 d^2-16 b c e d+3 b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 e d^2+24 b^2 c e^2 d-b^3 e^3\right )\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {\left (2 A c e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+24 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}+\frac {\left (B d \left (-7 b^3 e^3+56 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {\left (B d \left (-7 b^3 e^3+56 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {2 \left (2 A c e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+24 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {\left (B d \left (-7 b^3 e^3+56 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (2 A c e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+24 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{\sqrt {c} e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (2 A c e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+24 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{\sqrt {c} e}-\frac {2 \left (B d \left (-7 b^3 e^3+56 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )\right ) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (2 A c e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+24 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{\sqrt {c} e}+\frac {\left (B d \left (-7 b^3 e^3+56 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}}{2 e^2}-\frac {\sqrt {b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e^2}\) |
((2*B*d - A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(3*e^2*(d + e*x)^3) - (5*(((4* B*d*(2*c*d - b*e) - A*e*(4*c*d - b*e) + e*(4*B*c*d - b*B*e - 2*A*c*e)*x)*( b*x + c*x^2)^(3/2))/(2*e^2*(d + e*x)^2) - (3*(-(((A*e*(16*c^2*d^2 - 12*b*c *d*e + b^2*e^2) - 2*B*d*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) + e*(4*A*c*e *(2*c*d - b*e) - B*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^ 2])/(e^2*(d + e*x))) + ((2*(2*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) - B*(64*c^3*d^3 - 80*b*c^2*d^2*e + 24*b^2*c*d*e^2 - b^3*e^3))*ArcTanh[(Sqr t[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e) + ((B*d*(64*c^3*d^3 - 112*b*c^2*d^ 2*e + 56*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18* b^2*c*d*e^2 - b^3*e^3))*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*Sqrt[c*d - b*e]))/(2*e^2)))/(4*e^ 2)))/(6*e^2)
3.12.89.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 1.12 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.92
| method | result | size |
| pseudoelliptic | \(\frac {\frac {45 \left (-\frac {8 \left (A e -\frac {7 B d}{3}\right ) d^{2} e b \,c^{\frac {5}{2}}}{3}+b^{2} d \,e^{2} \left (A e -\frac {28 B d}{9}\right ) c^{\frac {3}{2}}+\frac {16 d^{3} \left (A e -2 B d \right ) c^{\frac {7}{2}}}{9}-\frac {b^{3} e^{3} \sqrt {c}\, \left (A e -7 B d \right )}{18}\right ) \left (e x +d \right )^{3} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{4}+\frac {15 \left (\frac {\left (e x +d \right )^{3} \left (b^{2} \left (A c +\frac {B b}{6}\right ) e^{3}-\frac {16 c d \left (A c +\frac {3 B b}{4}\right ) b \,e^{2}}{3}+\frac {16 c^{2} \left (A c +\frac {5 B b}{2}\right ) d^{2} e}{3}-\frac {32 B \,c^{3} d^{3}}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{2}+\sqrt {x \left (c x +b \right )}\, e \left (\frac {\left (\frac {x^{4} \left (\frac {2 B x}{3}+A \right ) e^{5}}{5}-x^{3} d \left (\frac {2 B x}{5}+A \right ) e^{4}-\frac {22 x^{2} d^{2} \left (-\frac {3 B x}{11}+A \right ) e^{3}}{3}-10 x \,d^{3} \left (-\frac {22 B x}{15}+A \right ) e^{2}-4 d^{4} \left (-5 B x +A \right ) e +8 B \,d^{5}\right ) c^{\frac {5}{2}}}{3}+\left (\left (\frac {3 x^{3} \left (\frac {13 B x}{27}+A \right ) e^{4}}{10}+\frac {35 \left (-\frac {69 B x}{175}+A \right ) x^{2} d \,e^{3}}{18}+\frac {23 d^{2} x \left (-2 B x +A \right ) e^{2}}{9}+d^{3} \left (A -\frac {61 B x}{9}\right ) e -\frac {8 B \,d^{4}}{3}\right ) c^{\frac {3}{2}}-\frac {\sqrt {c}\, \left (\frac {11 x^{2} \left (-B x +A \right ) e^{3}}{5}+\frac {8 x d \left (-\frac {93 B x}{20}+A \right ) e^{2}}{3}+d^{2} \left (-\frac {47 B x}{3}+A \right ) e -6 B \,d^{3}\right ) e b}{12}\right ) e b \right )\right ) \sqrt {d \left (b e -c d \right )}}{2}}{\sqrt {d \left (b e -c d \right )}\, e^{7} \left (e x +d \right )^{3} \sqrt {c}}\) | \(455\) |
| risch | \(\text {Expression too large to display}\) | \(2213\) |
| default | \(\text {Expression too large to display}\) | \(6565\) |
15/2/(d*(b*e-c*d))^(1/2)*(3/2*(-8/3*(A*e-7/3*B*d)*d^2*e*b*c^(5/2)+b^2*d*e^ 2*(A*e-28/9*B*d)*c^(3/2)+16/9*d^3*(A*e-2*B*d)*c^(7/2)-1/18*b^3*e^3*c^(1/2) *(A*e-7*B*d))*(e*x+d)^3*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+ (1/2*(e*x+d)^3*(b^2*(A*c+1/6*B*b)*e^3-16/3*c*d*(A*c+3/4*B*b)*b*e^2+16/3*c^ 2*(A*c+5/2*B*b)*d^2*e-32/3*B*c^3*d^3)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2)) +(x*(c*x+b))^(1/2)*e*(1/3*(1/5*x^4*(2/3*B*x+A)*e^5-x^3*d*(2/5*B*x+A)*e^4-2 2/3*x^2*d^2*(-3/11*B*x+A)*e^3-10*x*d^3*(-22/15*B*x+A)*e^2-4*d^4*(-5*B*x+A) *e+8*B*d^5)*c^(5/2)+((3/10*x^3*(13/27*B*x+A)*e^4+35/18*(-69/175*B*x+A)*x^2 *d*e^3+23/9*d^2*x*(-2*B*x+A)*e^2+d^3*(A-61/9*B*x)*e-8/3*B*d^4)*c^(3/2)-1/1 2*c^(1/2)*(11/5*x^2*(-B*x+A)*e^3+8/3*x*d*(-93/20*B*x+A)*e^2+d^2*(-47/3*B*x +A)*e-6*B*d^3)*e*b)*e*b))*(d*(b*e-c*d))^(1/2))/c^(1/2)/(e*x+d)^3/e^7
Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (463) = 926\).
Time = 3.15 (sec) , antiderivative size = 5934, normalized size of antiderivative = 11.99 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\left (d + e x\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1929 vs. \(2 (463) = 926\).
Time = 0.44 (sec) , antiderivative size = 1929, normalized size of antiderivative = 3.90 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Too large to display} \]
1/24*sqrt(c*x^2 + b*x)*(2*x*(4*B*c^2*x/e^4 - (24*B*c^4*d*e^17 - 13*B*b*c^3 *e^18 - 6*A*c^4*e^18)/(c^2*e^22)) + 3*(80*B*c^4*d^2*e^16 - 72*B*b*c^3*d*e^ 17 - 32*A*c^4*d*e^17 + 11*B*b^2*c^2*e^18 + 18*A*b*c^3*e^18)/(c^2*e^22)) + 5/8*(64*B*c^3*d^4 - 112*B*b*c^2*d^3*e - 32*A*c^3*d^3*e + 56*B*b^2*c*d^2*e^ 2 + 48*A*b*c^2*d^2*e^2 - 7*B*b^3*d*e^3 - 18*A*b^2*c*d*e^3 + A*b^3*e^4)*arc tan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e)) /(sqrt(-c*d^2 + b*d*e)*e^7) + 5/16*(64*B*c^3*d^3 - 80*B*b*c^2*d^2*e - 32*A *c^3*d^2*e + 24*B*b^2*c*d*e^2 + 32*A*b*c^2*d*e^2 - B*b^3*e^3 - 6*A*b^2*c*e ^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/(sqrt(c)*e^7) + 1/24*(720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^3*d^4*e^2 - 1200*(sqrt( c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^2*d^3*e^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^3*d^3*e^3 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c*d^ 2*e^4 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b*c^2*d^2*e^4 - 87*(sqrt(c )*x - sqrt(c*x^2 + b*x))^5*B*b^3*d*e^5 - 306*(sqrt(c)*x - sqrt(c*x^2 + b*x ))^5*A*b^2*c*d*e^5 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^3*e^6 + 2592 *(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*c^(7/2)*d^5*e - 3840*(sqrt(c)*x - sqr t(c*x^2 + b*x))^4*B*b*c^(5/2)*d^4*e^2 - 1680*(sqrt(c)*x - sqrt(c*x^2 + b*x ))^4*A*c^(7/2)*d^4*e^2 + 1560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c^(3 /2)*d^3*e^3 + 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b*c^(5/2)*d^3*e^3 - 147*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*sqrt(c)*d^2*e^4 - 666*(sqr...
Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]