Integrand size = 28, antiderivative size = 516 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c d e+15 b^2 e^2\right )\right )-c e (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x\right ) \sqrt {b x+c x^2}}{15 d e^4 (c d-b e) \sqrt {d+e x}}-\frac {2 \left (d^2 (8 B c d-5 b B e-3 A c e)+e (B d (11 c d-8 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac {2 \sqrt {-b} \sqrt {c} \left (3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c d e+43 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d e^5 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \left (24 A c e (2 c d-b e)-B \left (128 c^2 d^2-104 b c d e+15 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/15*(d^2*(-3*A*c*e-5*B*b*e+8*B*c*d)+e*(B*d*(-8*b*e+11*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)^(5/2)+2/15*(3*A*e*(b^ 2*e^2-16*b*c*d*e+16*c^2*d^2)-B*d*(43*b^2*e^2-168*b*c*d*e+128*c^2*d^2))*Ell ipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1 /2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/d/e^5/(-b*e+c*d)/(1+e*x/d)^(1/2)/(c*x^2+ b*x)^(1/2)-2/15*(24*A*c*e*(-b*e+2*c*d)-B*(15*b^2*e^2-104*b*c*d*e+128*c^2*d ^2))*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1 /2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/e^5/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^ (1/2)-2/15*(d*(3*A*c*e*(-7*b*e+8*c*d)-B*(15*b^2*e^2-76*b*c*d*e+64*c^2*d^2) )-c*e*(B*d*(-13*b*e+16*c*d)-3*A*e*(-b*e+2*c*d))*x)*(c*x^2+b*x)^(1/2)/d/e^4 /(-b*e+c*d)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 24.03 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {2 (x (b+c x))^{3/2} \left (\sqrt {\frac {b}{c}} e x (b+c x) \left (3 d^2 (B d-A e) (c d-b e)^2-d (c d-b e) (B d (17 c d-11 b e)+6 A e (-2 c d+b e)) (d+e x)+\left (-3 A e \left (11 c^2 d^2-11 b c d e+b^2 e^2\right )+B d \left (73 c^2 d^2-93 b c d e+23 b^2 e^2\right )\right ) (d+e x)^2+5 B c d (c d-b e) (d+e x)^3\right )+(d+e x)^2 \left (\sqrt {\frac {b}{c}} \left (B d \left (-128 c^2 d^2+168 b c d e-43 b^2 e^2\right )+3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )\right ) (b+c x) (d+e x)+i b e \left (B d \left (-128 c^2 d^2+168 b c d e-43 b^2 e^2\right )+3 A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e (c d-b e) (4 B d (16 c d-7 b e)+3 A e (-8 c d+b e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{15 \sqrt {\frac {b}{c}} d e^5 (c d-b e) x^2 (b+c x)^2 (d+e x)^{5/2}} \]
(2*(x*(b + c*x))^(3/2)*(Sqrt[b/c]*e*x*(b + c*x)*(3*d^2*(B*d - A*e)*(c*d - b*e)^2 - d*(c*d - b*e)*(B*d*(17*c*d - 11*b*e) + 6*A*e*(-2*c*d + b*e))*(d + e*x) + (-3*A*e*(11*c^2*d^2 - 11*b*c*d*e + b^2*e^2) + B*d*(73*c^2*d^2 - 93 *b*c*d*e + 23*b^2*e^2))*(d + e*x)^2 + 5*B*c*d*(c*d - b*e)*(d + e*x)^3) + ( d + e*x)^2*(Sqrt[b/c]*(B*d*(-128*c^2*d^2 + 168*b*c*d*e - 43*b^2*e^2) + 3*A *e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2))*(b + c*x)*(d + e*x) + I*b*e*(B*d*( -128*c^2*d^2 + 168*b*c*d*e - 43*b^2*e^2) + 3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSin h[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(c*d - b*e)*(4*B*d*(16*c*d - 7* b*e) + 3*A*e*(-8*c*d + b*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*E llipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*Sqrt[b/c]*d*e^5 *(c*d - b*e)*x^2*(b + c*x)^2*(d + e*x)^(5/2))
Time = 0.88 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1229, 27, 1230, 27, 1269, 1169, 122, 120, 127, 126}
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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle -\frac {2 \int -\frac {(b d (8 B c d-5 b B e-3 A c e)+c (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x) \sqrt {c x^2+b x}}{2 (d+e x)^{3/2}}dx}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(b d (8 B c d-5 b B e-3 A c e)+c (B d (16 c d-13 b e)-3 A e (2 c d-b e)) x) \sqrt {c x^2+b x}}{(d+e x)^{3/2}}dx}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c e d+15 b^2 e^2\right )\right )+c \left (3 A e \left (16 c^2 d^2-16 b c e d+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c e d+43 b^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {b d \left (3 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-76 b c e d+15 b^2 e^2\right )\right )+c \left (3 A e \left (16 c^2 d^2-16 b c e d+b^2 e^2\right )-B d \left (128 c^2 d^2-168 b c e d+43 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\frac {c \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {\frac {\frac {c \sqrt {x} \sqrt {b+c x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {\frac {\frac {c \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}-\frac {2 \sqrt {b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{3 e^2 \sqrt {d+e x}}}{5 d e^2 (c d-b e)}-\frac {2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)}\) |
(-2*(d^2*(8*B*c*d - 5*b*B*e - 3*A*c*e) + e*(B*d*(11*c*d - 8*b*e) - 3*A*e*( 2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(15*d*e^2*(c*d - b*e)*(d + e*x)^(5/2 )) + ((-2*(d*(3*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 76*b*c*d*e + 15*b^ 2*e^2)) - c*e*(B*d*(16*c*d - 13*b*e) - 3*A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) + ((2*Sqrt[-b]*Sqrt[c]*(3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*d*(128*c^2*d^2 - 168*b*c*d*e + 43*b^2*e^2))*Sqrt [x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqr t[-b]], (b*e)/(c*d)])/(e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b ]*d*(c*d - b*e)*(24*A*c*e*(2*c*d - b*e) - B*(128*c^2*d^2 - 104*b*c*d*e + 1 5*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[( Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b* x + c*x^2]))/(3*e^2))/(5*d*e^2*(c*d - b*e))
3.13.63.3.1 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[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
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[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(939\) vs. \(2(462)=924\).
Time = 1.20 (sec) , antiderivative size = 940, normalized size of antiderivative = 1.82
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{7} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (6 A b \,e^{2}-12 A c d e -11 B b d e +17 B c \,d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{15 e^{6} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (3 A \,b^{2} e^{3}-33 A b c d \,e^{2}+33 A \,c^{2} d^{2} e -23 B \,b^{2} d \,e^{2}+93 B b c \,d^{2} e -73 B \,c^{2} d^{3}\right )}{15 d \left (b e -c d \right ) e^{5} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 B c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{4}}+\frac {2 \left (\frac {2 A b c \,e^{2}-3 A \,c^{2} d e +B \,b^{2} e^{2}-6 B b c d e +6 B \,c^{2} d^{2}}{e^{5}}-\frac {c \left (6 A b \,e^{2}-12 A c d e -11 B b d e +17 B c \,d^{2}\right )}{15 e^{5}}+\frac {3 A \,b^{2} e^{3}-33 A b c d \,e^{2}+33 A \,c^{2} d^{2} e -23 B \,b^{2} d \,e^{2}+93 B b c \,d^{2} e -73 B \,c^{2} d^{3}}{15 e^{5} d}-\frac {b \left (3 A \,b^{2} e^{3}-33 A b c d \,e^{2}+33 A \,c^{2} d^{2} e -23 B \,b^{2} d \,e^{2}+93 B b c \,d^{2} e -73 B \,c^{2} d^{3}\right )}{15 e^{4} d \left (b e -c d \right )}-\frac {B c b d}{3 e^{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 \left (A c e +2 B b e -3 B c d \right )}{e^{4}}-\frac {c \left (3 A \,b^{2} e^{3}-33 A b c d \,e^{2}+33 A \,c^{2} d^{2} e -23 B \,b^{2} d \,e^{2}+93 B b c \,d^{2} e -73 B \,c^{2} d^{3}\right )}{15 e^{4} d \left (b e -c d \right )}-\frac {2 B c \left (b e +c d \right )}{3 e^{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 {e x +d}\, x \left (c x +b \right )}\) | \(940\) |
| default | \(\text {Expression too large to display}\) | \(4120\) |
1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((e*x+d)*x*(c*x+b))^(1/2)/x/(c*x+b)*(2/5 *d*(A*b*e^2-A*c*d*e-B*b*d*e+B*c*d^2)/e^7*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^( 1/2)/(x+d/e)^3-2/15*(6*A*b*e^2-12*A*c*d*e-11*B*b*d*e+17*B*c*d^2)/e^6*(c*e* x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^2+2/15*(c*e*x^2+b*e*x)/d/(b*e-c*d )/e^5*(3*A*b^2*e^3-33*A*b*c*d*e^2+33*A*c^2*d^2*e-23*B*b^2*d*e^2+93*B*b*c*d ^2*e-73*B*c^2*d^3)/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2/3*B*c/e^4*(c*e*x^3+b* e*x^2+c*d*x^2+b*d*x)^(1/2)+2*((2*A*b*c*e^2-3*A*c^2*d*e+B*b^2*e^2-6*B*b*c*d *e+6*B*c^2*d^2)/e^5-1/15*c*(6*A*b*e^2-12*A*c*d*e-11*B*b*d*e+17*B*c*d^2)/e^ 5+1/15/e^5*(3*A*b^2*e^3-33*A*b*c*d*e^2+33*A*c^2*d^2*e-23*B*b^2*d*e^2+93*B* b*c*d^2*e-73*B*c^2*d^3)/d-1/15*b/e^4/d/(b*e-c*d)*(3*A*b^2*e^3-33*A*b*c*d*e ^2+33*A*c^2*d^2*e-23*B*b^2*d*e^2+93*B*b*c*d^2*e-73*B*c^2*d^3)-1/3*B*c/e^4* b*d)*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*(c/e^4*(A*c*e+2*B*b*e-3*B*c*d)-1/15/e^4*c*(3*A*b^2*e^3-3 3*A*b*c*d*e^2+33*A*c^2*d^2*e-23*B*b^2*d*e^2+93*B*b*c*d^2*e-73*B*c^2*d^3)/d /(b*e-c*d)-2/3*B*c/e^4*(b*e+c*d))*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 = 0.20 (sec) , antiderivative size = 1252, normalized size of antiderivative = 2.43 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \]
2/45*((128*B*c^3*d^7 - 3*A*b^3*d^3*e^4 - 8*(29*B*b*c^2 + 6*A*c^3)*d^6*e + (103*B*b^2*c + 72*A*b*c^2)*d^5*e^2 - 2*(B*b^3 + 9*A*b^2*c)*d^4*e^3 + (128* B*c^3*d^4*e^3 - 3*A*b^3*e^7 - 8*(29*B*b*c^2 + 6*A*c^3)*d^3*e^4 + (103*B*b^ 2*c + 72*A*b*c^2)*d^2*e^5 - 2*(B*b^3 + 9*A*b^2*c)*d*e^6)*x^3 + 3*(128*B*c^ 3*d^5*e^2 - 3*A*b^3*d*e^6 - 8*(29*B*b*c^2 + 6*A*c^3)*d^4*e^3 + (103*B*b^2* c + 72*A*b*c^2)*d^3*e^4 - 2*(B*b^3 + 9*A*b^2*c)*d^2*e^5)*x^2 + 3*(128*B*c^ 3*d^6*e - 3*A*b^3*d^2*e^5 - 8*(29*B*b*c^2 + 6*A*c^3)*d^5*e^2 + (103*B*b^2* c + 72*A*b*c^2)*d^4*e^3 - 2*(B*b^3 + 9*A*b^2*c)*d^3*e^4)*x)*sqrt(c*e)*weie rstrassPInverse(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*(128*B*c^3*d^6*e - 3*A*b^2*c*d^3*e^4 - 24*(7*B*b*c^ 2 + 2*A*c^3)*d^5*e^2 + (43*B*b^2*c + 48*A*b*c^2)*d^4*e^3 + (128*B*c^3*d^3* e^4 - 3*A*b^2*c*e^7 - 24*(7*B*b*c^2 + 2*A*c^3)*d^2*e^5 + (43*B*b^2*c + 48* A*b*c^2)*d*e^6)*x^3 + 3*(128*B*c^3*d^4*e^3 - 3*A*b^2*c*d*e^6 - 24*(7*B*b*c ^2 + 2*A*c^3)*d^3*e^4 + (43*B*b^2*c + 48*A*b*c^2)*d^2*e^5)*x^2 + 3*(128*B* c^3*d^5*e^2 - 3*A*b^2*c*d^2*e^5 - 24*(7*B*b*c^2 + 2*A*c^3)*d^4*e^3 + (43*B *b^2*c + 48*A*b*c^2)*d^3*e^4)*x)*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 + ...
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]