Integrand size = 23, antiderivative size = 457 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-240 b c^2 d^2 e+111 b^2 c d e^2-b^3 e^3-3 c e \left (32 c^2 d^2-32 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \sqrt {-b} \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) (2 c d-b e) \left (128 c^2 d^2-128 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2*(c*x^2+b*x)^(5/2)/e/(e*x+d)^(1/2)-10/63*(-14*c*e*x-15*b*e+16*c*d)*(c*x^ 2+b*x)^(3/2)*(e*x+d)^(1/2)/e^3+4/63*(-b^4*e^4-7*b^3*c*d*e^3+135*b^2*c^2*d^ 2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)*EllipticE(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)*(e*x+d)^(1/2)/c^(3/2)/e^ 6/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/63*d*(-b*e+c*d)*(-b*e+2*c*d)*(-b^2*e ^2-128*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)/c^(3/2)/e^6/(e* x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/63*(128*c^3*d^3-240*b*c^2*d^2*e+111*b^2*c*d *e^2-b^3*e^3-3*c*e*(b^2*e^2-32*b*c*d*e+32*c^2*d^2)*x)*(e*x+d)^(1/2)*(c*x^2 +b*x)^(1/2)/c/e^5
Result contains complex when optimal does not.
Time = 24.49 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.09 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 (x (b+c x))^{5/2} \left (\frac {2 \left (128 c^4 d^4-256 b c^3 d^3 e+135 b^2 c^2 d^2 e^2-7 b^3 c d e^3-b^4 e^4\right ) (b+c x) (d+e x)}{c \sqrt {x}}-e \sqrt {x} (b+c x) \left (-b^3 e^3 (d+e x)+3 b^2 c e^2 \left (37 d^2+11 d e x-5 e^2 x^2\right )-b c^2 e \left (240 d^3+64 d^2 e x-31 d e^2 x^2+19 e^3 x^3\right )+c^3 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )-2 i \sqrt {\frac {b}{c}} e \left (-128 c^4 d^4+256 b c^3 d^3 e-135 b^2 c^2 d^2 e^2+7 b^3 c d e^3+b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \sqrt {\frac {b}{c}} e \left (-128 c^4 d^4+272 b c^3 d^3 e-159 b^2 c^2 d^2 e^2+13 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{63 c e^6 x^{5/2} (b+c x)^3 \sqrt {d+e x}} \]
(2*(x*(b + c*x))^(5/2)*((2*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^ 2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*(b + c*x)*(d + e*x))/(c*Sqrt[x]) - e*Sqrt [x]*(b + c*x)*(-(b^3*e^3*(d + e*x)) + 3*b^2*c*e^2*(37*d^2 + 11*d*e*x - 5*e ^2*x^2) - b*c^2*e*(240*d^3 + 64*d^2*e*x - 31*d*e^2*x^2 + 19*e^3*x^3) + c^3 *(128*d^4 + 32*d^3*e*x - 16*d^2*e^2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)) - (2* I)*Sqrt[b/c]*e*(-128*c^4*d^4 + 256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 + 7*b ^3*c*d*e^3 + b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*Ar cSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*e*(-128*c^4*d^4 + 272 *b*c^3*d^3*e - 159*b^2*c^2*d^2*e^2 + 13*b^3*c*d*e^3 + 2*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d) /(b*e)]))/(63*c*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])
Time = 0.74 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1161, 1231, 27, 1231, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x\right )^{3/2}}{\sqrt {d+e x}}dx}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (-\frac {2 \int -\frac {c \left (b d (16 c d-15 b e)+\left (32 c^2 d^2-32 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{2 \sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {\left (b d (16 c d-15 b e)+\left (32 c^2 d^2-32 b c e d+b^2 e^2\right ) x\right ) \sqrt {c x^2+b x}}{\sqrt {d+e x}}dx}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {5 \left (\frac {-\frac {2 \int -\frac {b d \left (128 c^3 d^3-240 b c^2 e d^2+111 b^2 c e^2 d-b^3 e^3\right )+2 \left (128 c^4 d^4-256 b c^3 e d^3+135 b^2 c^2 e^2 d^2-7 b^3 c e^3 d-b^4 e^4\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\int \frac {b d \left (128 c^3 d^3-240 b c^2 e d^2+111 b^2 c e^2 d-b^3 e^3\right )+2 \left (128 c^4 d^4-256 b c^3 e d^3+135 b^2 c^2 e^2 d^2-7 b^3 c e^3 d-b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {2 \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {2 \sqrt {x} \sqrt {b+c x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{15 c e^2}}{21 e^2}-\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (-15 b e+16 c d-14 c e x)}{63 e^2}\right )}{e}-\frac {2 \left (b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
(-2*(b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (5*((-2*Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*x)*(b*x + c*x^2)^(3/2))/(63*e^2) + ((-2*Sqrt[d + e*x]*( 128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3 - 3*c*e*(32*c^2* d^2 - 32*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(15*c*e^2) + ((4*Sqrt[-b ]*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b ^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]* Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d *e - 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]))/(15*c*e^2))/(21*e^2)))/e
3.5.1.3.1 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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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. \(1169\) vs. \(2(399)=798\).
Time = 2.09 (sec) , antiderivative size = 1170, normalized size of antiderivative = 2.56
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1170\) |
| elliptic | \(\text {Expression too large to display}\) | \(1532\) |
2/63*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(26*b*c^5*e^5*x^5-10*c^6*d*e^4*x^5+34 *b^2*c^4*e^5*x^4+16*c^6*d^2*e^3*x^4+16*b^3*c^3*e^5*x^3-32*c^6*d^3*e^2*x^3+ b^4*c^2*e^5*x^2-128*c^6*d^4*e*x^2+7*c^6*e^5*x^6+((c*x+b)/b)^(1/2)*(-c*(e*x +d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e- c*d))^(1/2))*b^5*c*d*e^4+125*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2 )*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c^ 2*d^2*e^3-510*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2 )*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^3*d^3*e^2+640*( (c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c *x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^4*d^4*e+12*((c*x+b)/b)^(1/2)*( -c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b* e/(b*e-c*d))^(1/2))*b^5*c*d*e^4-284*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d ))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2)) *b^4*c^2*d^2*e^3+782*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/ b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^3*d^3*e^ 2-768*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt icE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^4*d^4*e-256*((c*x+b)/b) ^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^( 1/2),(b*e/(b*e-c*d))^(1/2))*b*c^5*d^5+256*((c*x+b)/b)^(1/2)*(-c*(e*x+d)/(b *e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 750, normalized size of antiderivative = 1.64 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left ({\left (256 \, c^{5} d^{6} - 640 \, b c^{4} d^{5} e + 478 \, b^{2} c^{3} d^{4} e^{2} - 77 \, b^{3} c^{2} d^{3} e^{3} - 13 \, b^{4} c d^{2} e^{4} - 2 \, b^{5} d e^{5} + {\left (256 \, c^{5} d^{5} e - 640 \, b c^{4} d^{4} e^{2} + 478 \, b^{2} c^{3} d^{3} e^{3} - 77 \, b^{3} c^{2} d^{2} e^{4} - 13 \, b^{4} c d e^{5} - 2 \, b^{5} e^{6}\right )} x\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 ) + 6 \, {\left (128 \, c^{5} d^{5} e - 256 \, b c^{4} d^{4} e^{2} + 135 \, b^{2} c^{3} d^{3} e^{3} - 7 \, b^{3} c^{2} d^{2} e^{4} - b^{4} c d e^{5} + {\left (128 \, c^{5} d^{4} e^{2} - 256 \, b c^{4} d^{3} e^{3} + 135 \, b^{2} c^{3} d^{2} e^{4} - 7 \, b^{3} c^{2} d e^{5} - b^{4} c e^{6}\right )} x\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 (7 \, c^{5} e^{6} x^{4} - 128 \, c^{5} d^{4} e^{2} + 240 \, b c^{4} d^{3} e^{3} - 111 \, b^{2} c^{3} d^{2} e^{4} + b^{3} c^{2} d e^{5} - {\left (10 \, c^{5} d e^{5} - 19 \, b c^{4} e^{6}\right )} x^{3} + {\left (16 \, c^{5} d^{2} e^{4} - 31 \, b c^{4} d e^{5} + 15 \, b^{2} c^{3} e^{6}\right )} x^{2} - {\left (32 \, c^{5} d^{3} e^{3} - 64 \, b c^{4} d^{2} e^{4} + 33 \, b^{2} c^{3} d e^{5} - b^{3} c^{2} e^{6}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{189 \, {\left (c^{3} e^{8} x + c^{3} d e^{7}\right )}} \]
-2/189*((256*c^5*d^6 - 640*b*c^4*d^5*e + 478*b^2*c^3*d^4*e^2 - 77*b^3*c^2* d^3*e^3 - 13*b^4*c*d^2*e^4 - 2*b^5*d*e^5 + (256*c^5*d^5*e - 640*b*c^4*d^4* e^2 + 478*b^2*c^3*d^3*e^3 - 77*b^3*c^2*d^2*e^4 - 13*b^4*c*d*e^5 - 2*b^5*e^ 6)*x)*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)) + 6*(128*c^5*d^5*e - 256*b*c^4*d^4* e^2 + 135*b^2*c^3*d^3*e^3 - 7*b^3*c^2*d^2*e^4 - b^4*c*d*e^5 + (128*c^5*d^4 *e^2 - 256*b*c^4*d^3*e^3 + 135*b^2*c^3*d^2*e^4 - 7*b^3*c^2*d*e^5 - b^4*c*e ^6)*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 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3* c*e*x + c*d + b*e)/(c*e))) - 3*(7*c^5*e^6*x^4 - 128*c^5*d^4*e^2 + 240*b*c^ 4*d^3*e^3 - 111*b^2*c^3*d^2*e^4 + b^3*c^2*d*e^5 - (10*c^5*d*e^5 - 19*b*c^4 *e^6)*x^3 + (16*c^5*d^2*e^4 - 31*b*c^4*d*e^5 + 15*b^2*c^3*e^6)*x^2 - (32*c ^5*d^3*e^3 - 64*b*c^4*d^2*e^4 + 33*b^2*c^3*d*e^5 - b^3*c^2*e^6)*x)*sqrt(c* x^2 + b*x)*sqrt(e*x + d))/(c^3*e^8*x + c^3*d*e^7)
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]