Integrand size = 26, antiderivative size = 476 \[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\frac {2 \left (48 b^3 c^3-16 a b^2 c^2 d-9 a^2 b c d^2-8 a^3 d^3\right ) \sqrt {c+d x}}{105 a^3 c^3 e^4 \sqrt {e x} \sqrt {a+b x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{35 a^2 c e^2 (e x)^{5/2}}-\frac {2 \left (24 b^2 c^2-5 a b c d-4 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{105 a^3 c^2 e^3 (e x)^{3/2}}+\frac {2 \sqrt {b} \left (48 b^3 c^3-16 a b^2 c^2 d-9 a^2 b c d^2-8 a^3 d^3\right ) \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{105 a^{7/2} c^3 e^{9/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {2 \sqrt {b} d \left (24 b^2 c^2-5 a b c d-4 a^2 d^2\right ) \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{105 a^{5/2} c^3 e^{9/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:
2/105*(-8*a^3*d^3-9*a^2*b*c*d^2-16*a*b^2*c^2*d+48*b^3*c^3)*(d*x+c)^(1/2)/a ^3/c^3/e^4/(e*x)^(1/2)/(b*x+a)^(1/2)-2/7*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/e/( e*x)^(7/2)+2/35*(-a*d+6*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c/e^2/(e*x)^( 5/2)-2/105*(-4*a^2*d^2-5*a*b*c*d+24*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a ^3/c^2/e^3/(e*x)^(3/2)+2/105*b^(1/2)*(-8*a^3*d^3-9*a^2*b*c*d^2-16*a*b^2*c^ 2*d+48*b^3*c^3)*(d*x+c)^(1/2)*EllipticE(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2 )/(1+b*x/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(7/2)/c^3/e^(9/2)/(b*x+a)^(1/2)/(a* (d*x+c)/c/(b*x+a))^(1/2)-2/105*b^(1/2)*d*(-4*a^2*d^2-5*a*b*c*d+24*b^2*c^2) *(d*x+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2)) ,(1-a*d/b/c)^(1/2))/a^(5/2)/c^3/e^(9/2)/(b*x+a)^(1/2)/(a*(d*x+c)/c/(b*x+a) )^(1/2)
Result contains complex when optimal does not.
Time = 16.40 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\frac {\sqrt {e x} \left (2 a c (a+b x) (c+d x) \left (-24 b^2 c^2 x^2+a b c x (18 c+5 d x)+a^2 \left (-15 c^2-3 c d x+4 d^2 x^2\right )\right )+2 i \sqrt {\frac {a}{b}} b d \left (-48 b^3 c^3+16 a b^2 c^2 d+9 a^2 b c d^2+8 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{9/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} b d \left (-24 b^3 c^3+11 a b^2 c^2 d+5 a^2 b c d^2+8 a^3 d^3\right ) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{9/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )\right )}{105 a^4 c^3 e^5 x^4 \sqrt {a+b x} \sqrt {c+d x}} \] Input:
Integrate[Sqrt[c + d*x]/((e*x)^(9/2)*Sqrt[a + b*x]),x]
Output:
(Sqrt[e*x]*(2*a*c*(a + b*x)*(c + d*x)*(-24*b^2*c^2*x^2 + a*b*c*x*(18*c + 5 *d*x) + a^2*(-15*c^2 - 3*c*d*x + 4*d^2*x^2)) + (2*I)*Sqrt[a/b]*b*d*(-48*b^ 3*c^3 + 16*a*b^2*c^2*d + 9*a^2*b*c*d^2 + 8*a^3*d^3)*Sqrt[1 + a/(b*x)]*Sqrt [1 + c/(d*x)]*x^(9/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] - (2*I)*Sqrt[a/b]*b*d*(-24*b^3*c^3 + 11*a*b^2*c^2*d + 5*a^2*b*c*d^2 + 8*a ^3*d^3)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(9/2)*EllipticF[I*ArcSinh[Sq rt[a/b]/Sqrt[x]], (b*c)/(a*d)]))/(105*a^4*c^3*e^5*x^4*Sqrt[a + b*x]*Sqrt[c + d*x])
Time = 0.59 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {110, 27, 169, 27, 169, 27, 169, 27, 176, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \int -\frac {6 b c-a d+5 b d x}{2 (e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {6 b c-a d+5 b d x}{(e x)^{7/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {e \left (24 b^2 c^2-5 a b d c-4 a^2 d^2+3 b d (6 b c-a d) x\right )}{2 (e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{5 a c e^2}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {24 b^2 c^2-5 a b d c-4 a^2 d^2+3 b d (6 b c-a d) x}{(e x)^{5/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {e \left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3+b d \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right ) x\right )}{2 (e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c e^2}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3+b d \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right ) x}{(e x)^{3/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {-\frac {-\frac {-\frac {2 \int -\frac {b d e \left (a c \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right )+\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) x\right )}{2 \sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c e^2}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {b d \int \frac {a c \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right )+\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) x}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {b d \left (\frac {\left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e x} \sqrt {a+b x}}dx}{d}-\frac {4 c (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {b d \left (\frac {\sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {e x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {4 c (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {b d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {4 c (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {b d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {4 c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c e \sqrt {e x}}}{3 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle -\frac {-\frac {-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right )}{3 a c e (e x)^{3/2}}-\frac {\frac {b d \left (\frac {2 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {8 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (b c-a d) \left (a^2 d^2+2 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {-a} \sqrt {e}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {e} \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c e \sqrt {e x}}}{3 a c e}}{5 a c e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} (6 b c-a d)}{5 a c e (e x)^{5/2}}}{7 a e}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{7 a e (e x)^{7/2}}\) |
Input:
Int[Sqrt[c + d*x]/((e*x)^(9/2)*Sqrt[a + b*x]),x]
Output:
(-2*Sqrt[a + b*x]*Sqrt[c + d*x])/(7*a*e*(e*x)^(7/2)) - ((-2*(6*b*c - a*d)* Sqrt[a + b*x]*Sqrt[c + d*x])/(5*a*c*e*(e*x)^(5/2)) - ((-2*(24*b^2*c^2 - 5* a*b*c*d - 4*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(3*a*c*e*(e*x)^(3/2)) - ((-2*(48*b^3*c^3 - 16*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*Sqrt[a + b* x]*Sqrt[c + d*x])/(a*c*e*Sqrt[e*x]) + (b*d*((2*Sqrt[-a]*(48*b^3*c^3 - 16*a *b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*El lipticE[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sqr t[b]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (8*Sqrt[-a]*c*(b*c - a*d )*(12*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*E llipticF[ArcSin[(Sqrt[b]*Sqrt[e*x])/(Sqrt[-a]*Sqrt[e])], (a*d)/(b*c)])/(Sq rt[b]*d*Sqrt[e]*Sqrt[a + b*x]*Sqrt[c + d*x])))/(a*c*e))/(3*a*c*e))/(5*a*c* e))/(7*a*e)
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 5.00 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.63
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{7 e^{5} a \,x^{4}}-\frac {2 \left (a d -6 b c \right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{35 e^{5} a^{2} c \,x^{3}}+\frac {2 \left (4 a^{2} d^{2}+5 a b c d -24 b^{2} c^{2}\right ) \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}{105 e^{5} a^{3} c^{2} x^{2}}-\frac {2 \left (b d e \,x^{2}+a d e x +b c e x +a c e \right ) \left (8 a^{3} d^{3}+9 a^{2} b c \,d^{2}+16 a \,b^{2} c^{2} d -48 b^{3} c^{3}\right )}{105 e^{5} a^{4} c^{3} \sqrt {x \left (b d e \,x^{2}+a d e x +b c e x +a c e \right )}}+\frac {2 \left (\frac {b d \left (4 a^{2} d^{2}+5 a b c d -24 b^{2} c^{2}\right )}{105 e^{4} a^{3} c^{2}}-\frac {\left (a d +b c \right ) \left (8 a^{3} d^{3}+9 a^{2} b c \,d^{2}+16 a \,b^{2} c^{2} d -48 b^{3} c^{3}\right )}{105 a^{4} c^{3} e^{4}}+\frac {\left (a d e +b c e \right ) \left (8 a^{3} d^{3}+9 a^{2} b c \,d^{2}+16 a \,b^{2} c^{2} d -48 b^{3} c^{3}\right )}{105 e^{5} a^{4} c^{3}}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 b \left (8 a^{3} d^{3}+9 a^{2} b c \,d^{2}+16 a \,b^{2} c^{2} d -48 b^{3} c^{3}\right ) \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{105 a^{4} c^{2} e^{4} \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\) | \(777\) |
| default | \(\text {Expression too large to display}\) | \(1043\) |
Input:
int((d*x+c)^(1/2)/(e*x)^(9/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(e*x*(b*x+a)*(d*x+c))^(1/2)/(e*x)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*(-2/7/ e^5/a*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^4-2/35/e^5/a^2/c*(a* d-6*b*c)*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)/x^3+2/105/e^5/a^3/c ^2*(4*a^2*d^2+5*a*b*c*d-24*b^2*c^2)*(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x )^(1/2)/x^2-2/105*(b*d*e*x^2+a*d*e*x+b*c*e*x+a*c*e)/e^5/a^4/c^3*(8*a^3*d^3 +9*a^2*b*c*d^2+16*a*b^2*c^2*d-48*b^3*c^3)/(x*(b*d*e*x^2+a*d*e*x+b*c*e*x+a* c*e))^(1/2)+2*(1/105*b*d/e^4*(4*a^2*d^2+5*a*b*c*d-24*b^2*c^2)/a^3/c^2-1/10 5*(a*d+b*c)*(8*a^3*d^3+9*a^2*b*c*d^2+16*a*b^2*c^2*d-48*b^3*c^3)/a^4/c^3/e^ 4+1/105*(a*d*e+b*c*e)/e^5/a^4/c^3*(8*a^3*d^3+9*a^2*b*c*d^2+16*a*b^2*c^2*d- 48*b^3*c^3))*c/d*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)*(-1/c*x*d) ^(1/2)/(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)*EllipticF(((x+c/d)/c* d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))+2/105*b*(8*a^3*d^3+9*a^2*b*c*d^2+16*a*b^ 2*c^2*d-48*b^3*c^3)/a^4/c^2/e^4*((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^( 1/2)*(-1/c*x*d)^(1/2)/(b*d*e*x^3+a*d*e*x^2+b*c*e*x^2+a*c*e*x)^(1/2)*((-c/d +a/b)*EllipticE(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))-a/b*EllipticF (((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))))
Time = 0.09 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left ({\left (48 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d - 10 \, a^{2} b^{2} c^{2} d^{2} - 5 \, a^{3} b c d^{3} - 8 \, a^{4} d^{4}\right )} \sqrt {b d e} x^{4} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (48 \, b^{4} c^{3} d - 16 \, a b^{3} c^{2} d^{2} - 9 \, a^{2} b^{2} c d^{3} - 8 \, a^{3} b d^{4}\right )} \sqrt {b d e} x^{4} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) - 3 \, {\left (15 \, a^{3} b c^{3} d - {\left (48 \, b^{4} c^{3} d - 16 \, a b^{3} c^{2} d^{2} - 9 \, a^{2} b^{2} c d^{3} - 8 \, a^{3} b d^{4}\right )} x^{3} + {\left (24 \, a b^{3} c^{3} d - 5 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3}\right )} x^{2} - 3 \, {\left (6 \, a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x}\right )}}{315 \, a^{4} b c^{3} d e^{5} x^{4}} \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(9/2)/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/315*((48*b^4*c^4 - 40*a*b^3*c^3*d - 10*a^2*b^2*c^2*d^2 - 5*a^3*b*c*d^3 - 8*a^4*d^4)*sqrt(b*d*e)*x^4*weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a ^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^ 3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b*d)) + 3*(48*b^4*c^3*d - 16* a*b^3*c^2*d^2 - 9*a^2*b^2*c*d^3 - 8*a^3*b*d^4)*sqrt(b*d*e)*x^4*weierstrass Zeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b ^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), weierstrassPInverse(4/3*( b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b*d))) - 3*(15*a^3*b*c^3*d - (48*b^4*c^3*d - 16*a*b^3*c^2*d^2 - 9*a^2*b^2*c*d^3 - 8*a^3*b*d^4)*x^3 + (24*a*b^3*c^3*d - 5*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3)*x^ 2 - 3*(6*a^2*b^2*c^3*d - a^3*b*c^2*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqr t(e*x))/(a^4*b*c^3*d*e^5*x^4)
Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(1/2)/(e*x)**(9/2)/(b*x+a)**(1/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\int { \frac {\sqrt {d x + c}}{\sqrt {b x + a} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(9/2)/(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x + c)/(sqrt(b*x + a)*(e*x)^(9/2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\int { \frac {\sqrt {d x + c}}{\sqrt {b x + a} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((d*x+c)^(1/2)/(e*x)^(9/2)/(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x + c)/(sqrt(b*x + a)*(e*x)^(9/2)), x)
Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\int \frac {\sqrt {c+d\,x}}{{\left (e\,x\right )}^{9/2}\,\sqrt {a+b\,x}} \,d x \] Input:
int((c + d*x)^(1/2)/((e*x)^(9/2)*(a + b*x)^(1/2)),x)
Output:
int((c + d*x)^(1/2)/((e*x)^(9/2)*(a + b*x)^(1/2)), x)
\[ \int \frac {\sqrt {c+d x}}{(e x)^{9/2} \sqrt {a+b x}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a \,x^{4}+\sqrt {x}\, b \,x^{5}}d x \right )}{e^{5}} \] Input:
int((d*x+c)^(1/2)/(e*x)^(9/2)/(b*x+a)^(1/2),x)
Output:
(sqrt(e)*int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*a*x**4 + sqrt(x)*b*x** 5),x))/e**5