Integrand size = 28, antiderivative size = 410 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b \left (8 A c^2 d+b^2 B e-b c (4 B d+5 A e)\right )+c \left (16 A c^2 d+b^2 B e-8 b c (B d+A e)\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^2 d+b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \left (16 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (5 B d+3 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/3*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)*(e*x+d)^(1/2)/b^2/c/(c* x^2+b*x)^(3/2)+2/3*(b*(8*A*c^2*d+b^2*B*e-b*c*(5*A*e+4*B*d))+c*(16*A*c^2*d+ b^2*B*e-8*b*c*(A*e+B*d))*x)*(e*x+d)^(1/2)/b^4/c/(c*x^2+b*x)^(1/2)-2/3*(16* A*c^2*d+b^2*B*e-8*b*c*(A*e+B*d))*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e /c/d)^(1/2))*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/c^(1/2)/(1+e *x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*(16*A*c^2*d^2-8*b*c*d*(2*A*e+B*d)+b^2*e* (3*A*e+5*B*d))*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*x^(1/ 2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(7/2)/c^(1/2)/(e*x+d)^(1/2)/(c*x^2 +b*x)^(1/2)
Result contains complex when optimal does not.
Time = 22.76 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b (d+e x) \left (b B x \left (8 c^2 d x^2+b^2 (3 d-2 e x)+b c x (12 d-e x)\right )+A \left (-16 c^3 d x^3+8 b c^2 x^2 (-3 d+e x)+b^3 (d+4 e x)+b^2 c x (-6 d+13 e x)\right )\right )+\sqrt {\frac {b}{c}} x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^2 d+b^2 B e-8 b c (B d+A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^2 d+b^2 B e-8 b c (B d+A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 A c^2 d+b^2 B e-b c (4 B d+5 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 (x (b+c x))^{3/2} \sqrt {d+e x}} \]
(-2*(b*(d + e*x)*(b*B*x*(8*c^2*d*x^2 + b^2*(3*d - 2*e*x) + b*c*x*(12*d - e *x)) + A*(-16*c^3*d*x^3 + 8*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 4*e*x) + b^2 *c*x*(-6*d + 13*e*x))) + Sqrt[b/c]*x*(b + c*x)*(Sqrt[b/c]*(16*A*c^2*d + b^ 2*B*e - 8*b*c*(B*d + A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^2*d + b^2*B *e - 8*b*c*(B*d + A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellipt icE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*A*c^2*d + b^2*B* e - b*c*(4*B*d + 5*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellip ticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*(x*(b + c*x))^(3 /2)*Sqrt[d + e*x])
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int \frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d \left (-B e b^2+4 B c d b+5 A c e b-8 A c^2 d\right )+3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {d \left (B e b^2-c (4 B d+5 A e) b+8 A c^2 d\right )-3 c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}\) |
3.13.80.3.1 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[((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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(726\) vs. \(2(356)=712\).
Time = 1.22 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.77
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 d A \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (4 A b e -8 A c d +3 B b d \right )}{3 b^{4} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 \left (A b c e -A \,c^{2} d -b^{2} B e +B b c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c^{2} b^{3} \left (x +\frac {b}{c}\right )^{2}}-\frac {2 \left (c e \,x^{2}+c d x \right ) \left (4 A b c e -8 A \,c^{2} d -b^{2} B e +5 B b c d \right )}{3 c \,b^{4} \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {A c d e}{3 b^{3}}-\frac {\left (A b c e -A \,c^{2} d -b^{2} B e +B b c d \right ) e}{3 c \,b^{3}}+\frac {\left (4 A b c e -8 A \,c^{2} d -b^{2} B e +5 B b c d \right ) \left (b e -c d \right )}{3 c \,b^{4}}+\frac {d \left (4 A b c e -8 A \,c^{2} d -b^{2} B e +5 B b c d \right )}{3 b^{4}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {c e \left (4 A b e -8 A c d +3 B b d \right )}{3 b^{4}}+\frac {\left (4 A b c e -8 A \,c^{2} d -b^{2} B e +5 B b c d \right ) e}{3 b^{4}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(727\) |
| default | \(\text {Expression too large to display}\) | \(2044\) |
((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2/3*d*A/b^3*(c *e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/x^2-2/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4* (4*A*b*e-8*A*c*d+3*B*b*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2/3*(A*b*c*e -A*c^2*d-B*b^2*e+B*b*c*d)/c^2/b^3*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x +b/c)^2-2/3*(c*e*x^2+c*d*x)*(4*A*b*c*e-8*A*c^2*d-B*b^2*e+5*B*b*c*d)/c/b^4/ ((x+b/c)*(c*e*x^2+c*d*x))^(1/2)+2*(-1/3/b^3*A*c*d*e-1/3*(A*b*c*e-A*c^2*d-B *b^2*e+B*b*c*d)/c*e/b^3+1/3*(4*A*b*c*e-8*A*c^2*d-B*b^2*e+5*B*b*c*d)/c*(b*e -c*d)/b^4+1/3*d*(4*A*b*c*e-8*A*c^2*d-B*b^2*e+5*B*b*c*d)/b^4)*b/c*((x+b/c)/ b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d* x^2+b*d*x)^(1/2)*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))+2* (1/3*c*e*(4*A*b*e-8*A*c*d+3*B*b*d)/b^4+1/3*(4*A*b*c*e-8*A*c^2*d-B*b^2*e+5* B*b*c*d)*e/b^4)*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/2)*(-c*x/b )^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-b/c+d/e)*EllipticE(((x+b/ c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))-d/e*EllipticF(((x+b/c)/b*c)^(1/2),( -b/c/(-b/c+d/e))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 822, normalized size of antiderivative = 2.00 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} - {\left (5 \, B b^{2} c^{3} - 16 \, A b c^{4}\right )} d e - {\left (B b^{3} c^{2} + A b^{2} c^{3}\right )} e^{2}\right )} x^{4} + 2 \, {\left (8 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} - {\left (5 \, B b^{3} c^{2} - 16 \, A b^{2} c^{3}\right )} d e - {\left (B b^{4} c + A b^{3} c^{2}\right )} e^{2}\right )} x^{3} + {\left (8 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} - {\left (5 \, B b^{4} c - 16 \, A b^{3} c^{2}\right )} d e - {\left (B b^{5} + A b^{4} c\right )} e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d e - {\left (B b^{2} c^{3} - 8 \, A b c^{4}\right )} e^{2}\right )} x^{4} + 2 \, {\left (8 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d e - {\left (B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} e^{2}\right )} x^{3} + {\left (8 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d e - {\left (B b^{4} c - 8 \, A b^{3} c^{2}\right )} e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (A b^{3} c^{2} d e + {\left (8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d e - {\left (B b^{2} c^{3} - 8 \, A b c^{4}\right )} e^{2}\right )} x^{3} + {\left (12 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d e - {\left (2 \, B b^{3} c^{2} - 13 \, A b^{2} c^{3}\right )} e^{2}\right )} x^{2} + {\left (4 \, A b^{3} c^{2} e^{2} + 3 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d e\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{4} c^{4} e x^{4} + 2 \, b^{5} c^{3} e x^{3} + b^{6} c^{2} e x^{2}\right )}} \]
-2/9*(((8*(B*b*c^4 - 2*A*c^5)*d^2 - (5*B*b^2*c^3 - 16*A*b*c^4)*d*e - (B*b^ 3*c^2 + A*b^2*c^3)*e^2)*x^4 + 2*(8*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - (5*B*b^3* c^2 - 16*A*b^2*c^3)*d*e - (B*b^4*c + A*b^3*c^2)*e^2)*x^3 + (8*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - (5*B*b^4*c - 16*A*b^3*c^2)*d*e - (B*b^5 + A*b^4*c)*e^2 )*x^2)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^ 2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3 *e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*((8*(B*b*c^4 - 2*A*c^5)*d*e - (B*b^2*c^3 - 8*A*b*c^4)*e^2)*x^4 + 2*(8*(B*b^2*c^3 - 2*A*b*c^4)*d*e - (B*b ^3*c^2 - 8*A*b^2*c^3)*e^2)*x^3 + (8*(B*b^3*c^2 - 2*A*b^2*c^3)*d*e - (B*b^4 *c - 8*A*b^3*c^2)*e^2)*x^2)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d *e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e ^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^ 3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(A*b^3*c^2*d*e + (8*(B *b*c^4 - 2*A*c^5)*d*e - (B*b^2*c^3 - 8*A*b*c^4)*e^2)*x^3 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d*e - (2*B*b^3*c^2 - 13*A*b^2*c^3)*e^2)*x^2 + (4*A*b^3*c^2*e^ 2 + 3*(B*b^3*c^2 - 2*A*b^2*c^3)*d*e)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/( b^4*c^4*e*x^4 + 2*b^5*c^3*e*x^3 + b^6*c^2*e*x^2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]