Integrand size = 23, antiderivative size = 427 \[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\frac {4 \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d \left (8 b c^2-7 a d^2\right ) x\right ) \sqrt {a-b x^2}}{315 d^4}-\frac {2 (8 c-7 d x) \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}{63 d^2}-\frac {8 \sqrt {a} \left (32 b^2 c^4-57 a b c^2 d^2+21 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 \sqrt {b} d^5 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {8 \sqrt {a} c \left (32 b^2 c^4-65 a b c^2 d^2+33 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{315 \sqrt {b} d^5 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
4/315*(d*x+c)^(1/2)*(c*(-33*a*d^2+32*b*c^2)-3*d*(-7*a*d^2+8*b*c^2)*x)*(-b* x^2+a)^(1/2)/d^4-2/63*(-7*d*x+8*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(3/2)/d^2-8/31 5*a^(1/2)*(21*a^2*d^4-57*a*b*c^2*d^2+32*b^2*c^4)*(d*x+c)^(1/2)*(1-b*x^2/a) ^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)* d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^5/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^ (1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+8/315*a^(1/2)*c*(33*a^2*d^4-65*a*b*c^2*d^ 2+32*b^2*c^4)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1 /2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/( b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^5/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 13.07 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.43 \[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {2 (c+d x) \left (a d^2 (-106 c+77 d x)+b \left (64 c^3-48 c^2 d x+40 c d^2 x^2-35 d^3 x^3\right )\right )}{d^4}-\frac {8 \left (d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (32 b^2 c^4-57 a b c^2 d^2+21 a^2 d^4\right ) \left (a-b x^2\right )+i \sqrt {b} \left (32 b^{5/2} c^5-32 \sqrt {a} b^2 c^4 d-57 a b^{3/2} c^3 d^2+57 a^{3/2} b c^2 d^3+21 a^2 \sqrt {b} c d^4-21 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (32 b^2 c^4-8 \sqrt {a} b^{3/2} c^3 d-57 a b c^2 d^2+12 a^{3/2} \sqrt {b} c d^3+21 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{b d^6 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{315 \sqrt {c+d x}} \] Input:
Integrate[(x*(a - b*x^2)^(3/2))/Sqrt[c + d*x],x]
Output:
(Sqrt[a - b*x^2]*((2*(c + d*x)*(a*d^2*(-106*c + 77*d*x) + b*(64*c^3 - 48*c ^2*d*x + 40*c*d^2*x^2 - 35*d^3*x^3)))/d^4 - (8*(d^2*Sqrt[-c + (Sqrt[a]*d)/ Sqrt[b]]*(32*b^2*c^4 - 57*a*b*c^2*d^2 + 21*a^2*d^4)*(a - b*x^2) + I*Sqrt[b ]*(32*b^(5/2)*c^5 - 32*Sqrt[a]*b^2*c^4*d - 57*a*b^(3/2)*c^3*d^2 + 57*a^(3/ 2)*b*c^2*d^3 + 21*a^2*Sqrt[b]*c*d^4 - 21*a^(5/2)*d^5)*Sqrt[(d*(Sqrt[a]/Sqr t[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x ]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]*Sqrt[b]*d *(32*b^2*c^4 - 8*Sqrt[a]*b^(3/2)*c^3*d - 57*a*b*c^2*d^2 + 12*a^(3/2)*Sqrt[ b]*c*d^3 + 21*a^2*d^4)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-((( Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[ Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sq rt[b]*c - Sqrt[a]*d)]))/(b*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a - b*x^2)) ))/(315*Sqrt[c + d*x])
Time = 0.67 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {591, 27, 682, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 591 |
| \(\displaystyle \frac {4 \int -\frac {\left (a c d+\left (8 b c^2-7 a d^2\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {\left (a c d+\left (8 b c^2-7 a d^2\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {2 \left (-\frac {4 \int \frac {b \left (4 a c d \left (2 b c^2-3 a d^2\right )+\left (32 b^2 c^4-57 a b d^2 c^2+21 a^2 d^4\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \int \frac {4 a c d \left (2 b c^2-3 a d^2\right )+\left (32 b^2 c^4-57 a b d^2 c^2+21 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {\left (21 a^2 d^4-57 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (32 b c^2-33 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (21 a^2 d^4-57 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {c \left (32 b c^2-33 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 d^4-57 a b c^2 d^2+32 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {c \left (32 b c^2-33 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (-\frac {c \left (32 b c^2-33 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 d^4-57 a b c^2 d^2+32 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (32 b c^2-33 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 d^4-57 a b c^2 d^2+32 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (32 b c^2-33 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 d^4-57 a b c^2 d^2+32 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (32 b c^2-33 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 d^4-57 a b c^2 d^2+32 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}-\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (c \left (32 b c^2-33 a d^2\right )-3 d x \left (8 b c^2-7 a d^2\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} (8 c-7 d x) \sqrt {c+d x}}{63 d^2}\) |
Input:
Int[(x*(a - b*x^2)^(3/2))/Sqrt[c + d*x],x]
Output:
(-2*(8*c - 7*d*x)*Sqrt[c + d*x]*(a - b*x^2)^(3/2))/(63*d^2) - (2*((-2*Sqrt [c + d*x]*(c*(32*b*c^2 - 33*a*d^2) - 3*d*(8*b*c^2 - 7*a*d^2)*x)*Sqrt[a - b *x^2])/(15*d^2) - (2*((-2*Sqrt[a]*(32*b^2*c^4 - 57*a*b*c^2*d^2 + 21*a^2*d^ 4)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x) /Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqr t[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*( 32*b*c^2 - 33*a*d^2)*(b*c^2 - a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqr t[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x] *Sqrt[a - b*x^2])))/(15*d^2)))/(21*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(2*p + 1) - d*(n + 2*p + 1)*x)/ (d^2*(n + 2*p + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 2*p + 1)*(n + 2*p + 2))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*Simp[a*c*d*n + (b*c^2*(2*p + 1) + a*d^2*(n + 2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && LeQ[-1, n, 0] && !ILtQ[n + 2*p, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(715\) vs. \(2(357)=714\).
Time = 3.30 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.68
| method | result | size |
| risch | \(-\frac {2 \left (35 b \,d^{3} x^{3}-40 b c \,d^{2} x^{2}-77 a x \,d^{3}+48 b \,c^{2} d x +106 a \,d^{2} c -64 b \,c^{3}\right ) \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{315 d^{4}}-\frac {4 \left (-\frac {\left (21 a^{2} d^{4}-57 b \,c^{2} d^{2} a +32 b^{2} c^{4}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {12 d^{3} c \,a^{2} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {8 a \,c^{3} d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}}{315 d^{4} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(716\) |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 b \,x^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{9 d}+\frac {16 b c \,x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{63 d^{2}}-\frac {2 \left (-\frac {11 a b}{9}+\frac {16 b^{2} c^{2}}{21 d^{2}}\right ) x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (-\frac {4 c \left (-\frac {11 a b}{9}+\frac {16 b^{2} c^{2}}{21 d^{2}}\right )}{5 d}+\frac {2 a b c}{63 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (\frac {2 a c \left (-\frac {11 a b}{9}+\frac {16 b^{2} c^{2}}{21 d^{2}}\right )}{5 b d}+\frac {a \left (-\frac {4 c \left (-\frac {11 a b}{9}+\frac {16 b^{2} c^{2}}{21 d^{2}}\right )}{5 d}+\frac {2 a b c}{63 d}\right )}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (a^{2}-\frac {32 a b \,c^{2}}{63 d^{2}}+\frac {3 a \left (-\frac {11 a b}{9}+\frac {16 b^{2} c^{2}}{21 d^{2}}\right )}{5 b}-\frac {2 c \left (-\frac {4 c \left (-\frac {11 a b}{9}+\frac {16 b^{2} c^{2}}{21 d^{2}}\right )}{5 d}+\frac {2 a b c}{63 d}\right )}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(853\) |
| default | \(\text {Expression too large to display}\) | \(1651\) |
Input:
int(x*(-b*x^2+a)^(3/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/315*(35*b*d^3*x^3-40*b*c*d^2*x^2-77*a*d^3*x+48*b*c^2*d*x+106*a*c*d^2-64 *b*c^3)*(d*x+c)^(1/2)/d^4*(-b*x^2+a)^(1/2)-4/315/d^4*(-(21*a^2*d^4-57*a*b* c^2*d^2+32*b^2*c^4)/b*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/ 2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/ (a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((c/d-1/b*(a*b)^(1/2 ))*EllipticE(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*( a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c/d*EllipticF(1/2*2^(1/2)*((x+1/b *(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)) )^(1/2)))+12*d^3*c*a^2/b*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^ (1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2)) *b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1/2*2^( 1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b* (a*b)^(1/2)))^(1/2))-8*a*c^3*d*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/ (a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^ (1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(1 /2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/ d-1/b*(a*b)^(1/2)))^(1/2)))*((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x ^2+a)^(1/2)
Time = 0.10 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.73 \[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (4 \, {\left (32 \, b^{2} c^{5} - 81 \, a b c^{3} d^{2} + 57 \, a^{2} c d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 12 \, {\left (32 \, b^{2} c^{4} d - 57 \, a b c^{2} d^{3} + 21 \, a^{2} d^{5}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (35 \, b^{2} d^{5} x^{3} - 40 \, b^{2} c d^{4} x^{2} - 64 \, b^{2} c^{3} d^{2} + 106 \, a b c d^{4} + {\left (48 \, b^{2} c^{2} d^{3} - 77 \, a b d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{945 \, b d^{6}} \] Input:
integrate(x*(-b*x^2+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
2/945*(4*(32*b^2*c^5 - 81*a*b*c^3*d^2 + 57*a^2*c*d^4)*sqrt(-b*d)*weierstra ssPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3 ), 1/3*(3*d*x + c)/d) + 12*(32*b^2*c^4*d - 57*a*b*c^2*d^3 + 21*a^2*d^5)*sq rt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a *c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27* (b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) - 3*(35*b^2*d^5*x^3 - 40* b^2*c*d^4*x^2 - 64*b^2*c^3*d^2 + 106*a*b*c*d^4 + (48*b^2*c^2*d^3 - 77*a*b* d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b*d^6)
\[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\int \frac {x \left (a - b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x}}\, dx \] Input:
integrate(x*(-b*x**2+a)**(3/2)/(d*x+c)**(1/2),x)
Output:
Integral(x*(a - b*x**2)**(3/2)/sqrt(c + d*x), x)
\[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x*(-b*x^2+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate((-b*x^2 + a)^(3/2)*x/sqrt(d*x + c), x)
\[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{2}} x}{\sqrt {d x + c}} \,d x } \] Input:
integrate(x*(-b*x^2+a)^(3/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
integrate((-b*x^2 + a)^(3/2)*x/sqrt(d*x + c), x)
Timed out. \[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\int \frac {x\,{\left (a-b\,x^2\right )}^{3/2}}{\sqrt {c+d\,x}} \,d x \] Input:
int((x*(a - b*x^2)^(3/2))/(c + d*x)^(1/2),x)
Output:
int((x*(a - b*x^2)^(3/2))/(c + d*x)^(1/2), x)
\[ \int \frac {x \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}} \, dx=\frac {-\frac {4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a^{2} d^{3}}{15}+\frac {16 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b \,c^{2} d}{315}+\frac {22 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a b c \,d^{2} x}{45}-\frac {32 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{3} x}{105}+\frac {16 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c^{2} d \,x^{2}}{63}-\frac {2 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b^{2} c \,d^{2} x^{3}}{9}-\frac {2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,d^{4}}{5}+\frac {38 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{2} d^{2}}{35}-\frac {64 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c^{4}}{105}+\frac {2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{3} d^{4}}{15}-\frac {18 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} b \,c^{2} d^{2}}{35}+\frac {32 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a \,b^{2} c^{4}}{105}}{b c \,d^{3}} \] Input:
int(x*(-b*x^2+a)^(3/2)/(d*x+c)^(1/2),x)
Output:
(2*( - 42*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*d**3 + 8*sqrt(c + d*x)*sqrt( a - b*x**2)*a*b*c**2*d + 77*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c*d**2*x - 48*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**3*x + 40*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d*x**2 - 35*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c*d**2*x **3 - 63*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**2*b*d**4 + 171*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2 )/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**2*c**2*d**2 - 96*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**3 *c**4 + 21*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a**3*d**4 - 81*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a* d*x - b*c*x**2 - b*d*x**3),x)*a**2*b*c**2*d**2 + 48*int((sqrt(c + d*x)*sqr t(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b**2*c**4))/(315*b *c*d**3)