Integrand size = 27, antiderivative size = 383 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=-\frac {2 (3 B d+5 A e) \sqrt {d+e x} \sqrt {a-c x^2}}{15 c}-\frac {2 B (d+e x)^{3/2} \sqrt {a-c x^2}}{5 c}-\frac {2 \sqrt {a} \left (3 B c d^2+20 A c d e+9 a B e^2\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 c^{3/2} e \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {2 \sqrt {a} (3 B d+5 A e) \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 c^{3/2} e \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:
-2/15*(5*A*e+3*B*d)*(e*x+d)^(1/2)*(-c*x^2+a)^(1/2)/c-2/5*B*(e*x+d)^(3/2)*( -c*x^2+a)^(1/2)/c-2/15*a^(1/2)*(20*A*c*d*e+9*B*a*e^2+3*B*c*d^2)*(e*x+d)^(1 /2)*(1-c*x^2/a)^(1/2)*EllipticE(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^ (1/2)*(a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/c^(3/2)/e/(c^(1/2)*(e*x+d)/ (c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)+2/15*a^(1/2)*(5*A*e+3*B*d)*( -a*e^2+c*d^2)*(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1-c*x^2/a)^(1 /2)*EllipticF(1/2*(1-c^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/( c^(1/2)*d+a^(1/2)*e))^(1/2))/c^(3/2)/e/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.61 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=\frac {2 \sqrt {a-c x^2} \left (-3 B c d^2-20 A c d e-9 a B e^2-c (d+e x) (6 B d+5 A e+3 B e x)+\frac {i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (3 B c d^2+20 A c d e+9 a B e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}-\frac {i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (15 A c d+9 a B e-\sqrt {a} \sqrt {c} (3 B d+5 A e)\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{15 c^2 \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*(d + e*x)^(3/2))/Sqrt[a - c*x^2],x]
Output:
(2*Sqrt[a - c*x^2]*(-3*B*c*d^2 - 20*A*c*d*e - 9*a*B*e^2 - c*(d + e*x)*(6*B *d + 5*A*e + 3*B*e*x) + (I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(3*B*c*d^2 + 20 *A*c*d*e + 9*a*B*e^2)*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((S qrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[S qrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqr t[c]*d - Sqrt[a]*e)])/(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2)) - (I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(15*A*c*d + 9*a*B*e - Sqrt[a]*Sqrt[c]*( 3*B*d + 5*A*e))*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a] *e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)])/(e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(15*c^2* Sqrt[d + e*x])
Time = 0.58 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {687, 27, 687, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {2 \int -\frac {\sqrt {d+e x} (5 A c d+3 a B e+c (3 B d+5 A e) x)}{2 \sqrt {a-c x^2}}dx}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} (5 A c d+3 a B e+c (3 B d+5 A e) x)}{\sqrt {a-c x^2}}dx}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {c \left (12 a B d e+5 A \left (3 c d^2+a e^2\right )+\left (3 B c d^2+20 A c e d+9 a B e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{3 c}-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {15 A c d^2+12 a B e d+5 a A e^2+\left (3 B c d^2+20 A c e d+9 a B e^2\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {\left (9 a B e^2+20 A c d e+3 B c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) (5 A e+3 B d) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {\sqrt {1-\frac {c x^2}{a}} \left (9 a B e^2+20 A c d e+3 B c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) (5 A e+3 B d) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{3} \left (-\frac {\left (c d^2-a e^2\right ) (5 A e+3 B d) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2+20 A c d e+3 B c d^2\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{3} \left (-\frac {\left (c d^2-a e^2\right ) (5 A e+3 B d) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2+20 A c d e+3 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{3} \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (5 A e+3 B d) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2+20 A c d e+3 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (5 A e+3 B d) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2+20 A c d e+3 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (5 A e+3 B d) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (9 a B e^2+20 A c d e+3 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )-\frac {2}{3} \sqrt {a-c x^2} \sqrt {d+e x} (5 A e+3 B d)}{5 c}-\frac {2 B \sqrt {a-c x^2} (d+e x)^{3/2}}{5 c}\) |
Input:
Int[((A + B*x)*(d + e*x)^(3/2))/Sqrt[a - c*x^2],x]
Output:
(-2*B*(d + e*x)^(3/2)*Sqrt[a - c*x^2])/(5*c) + ((-2*(3*B*d + 5*A*e)*Sqrt[d + e*x]*Sqrt[a - c*x^2])/3 + ((-2*Sqrt[a]*(3*B*c*d^2 + 20*A*c*d*e + 9*a*B* e^2)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]* x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[(S qrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2*Sqrt[a]*( 3*B*d + 5*A*e)*(c*d^2 - a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[ a]*e)]*Sqrt[1 - (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/ Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[ a - c*x^2]))/3)/(5*c)
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[((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[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(681\) vs. \(2(313)=626\).
Time = 5.00 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.78
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 B e x \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{5 c}-\frac {2 \left (A \,e^{2}+\frac {6}{5} B d e \right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (A \,d^{2}+\frac {2 a d e B}{5 c}+\frac {a \left (A \,e^{2}+\frac {6}{5} B d e \right )}{3 c}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 \left (2 A d e +B \,d^{2}+\frac {3 a \,e^{2} B}{5 c}-\frac {2 d \left (A \,e^{2}+\frac {6}{5} B d e \right )}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(682\) |
| risch | \(-\frac {2 \left (3 B e x +5 A e +6 B d \right ) \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{15 c}+\frac {\left (\frac {\left (20 A c d e +9 B a \,e^{2}+3 B c \,d^{2}\right ) \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \left (\left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )-\frac {d \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{e}\right )}{c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {5 A a \,e^{2} \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {15 A \,d^{2} \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {12 B a d e \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right ) \sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}}{15 c \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\) | \(839\) |
| default | \(\text {Expression too large to display}\) | \(2022\) |
Input:
int((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(-2/5*B*e/c*x*(- c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)-2/3*(A*e^2+6/5*B*d*e)/c/e*(-c*e*x^3-c*d*x ^2+a*e*x+a*d)^(1/2)+2*(A*d^2+2/5*a/c*d*e*B+1/3*a/c*(A*e^2+6/5*B*d*e))*(d/e -1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2 ))/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2 )))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*( a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+ 2*(2*A*d*e+B*d^2+3/5*a/c*e^2*B-2/3*d/e*(A*e^2+6/5*B*d*e))*(d/e-1/c*(a*c)^( 1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c *(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(- c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-1/c*(a*c)^(1/2))*EllipticE(((x+d/e )/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/ 2)))^(1/2))+1/c*(a*c)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2 ),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))))
Time = 0.14 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.70 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=\frac {2 \, {\left ({\left (3 \, B c d^{3} - 25 \, A c d^{2} e - 27 \, B a d e^{2} - 15 \, A a e^{3}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (3 \, B c d^{2} e + 20 \, A c d e^{2} + 9 \, B a e^{3}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (3 \, B c e^{3} x + 6 \, B c d e^{2} + 5 \, A c e^{3}\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, c^{2} e^{2}} \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/45*((3*B*c*d^3 - 25*A*c*d^2*e - 27*B*a*d*e^2 - 15*A*a*e^3)*sqrt(-c*e)*we ierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2) /(c*e^3), 1/3*(3*e*x + d)/e) + 3*(3*B*c*d^2*e + 20*A*c*d*e^2 + 9*B*a*e^3)* sqrt(-c*e)*weierstrassZeta(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9 *a*d*e^2)/(c*e^3), weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/2 7*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e)) - 3*(3*B*c*e^3*x + 6*B* c*d*e^2 + 5*A*c*e^3)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(c^2*e^2)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\sqrt {a - c x^{2}}}\, dx \] Input:
integrate((B*x+A)*(e*x+d)**(3/2)/(-c*x**2+a)**(1/2),x)
Output:
Integral((A + B*x)*(d + e*x)**(3/2)/sqrt(a - c*x**2), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {-c x^{2} + a}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*(e*x + d)^(3/2)/sqrt(-c*x^2 + a), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {-c x^{2} + a}} \,d x } \] Input:
integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(e*x + d)^(3/2)/sqrt(-c*x^2 + a), x)
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {a-c\,x^2}} \,d x \] Input:
int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(1/2),x)
Output:
int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^(1/2), x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {a-c x^2}} \, dx=\frac {-6 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, a b \,e^{2}-20 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, a c d e -10 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, b c \,d^{2}-4 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, b c d e x -9 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a b c \,e^{3}-20 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a \,c^{2} d \,e^{2}-3 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) b \,c^{2} d^{2} e +3 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{2} b \,e^{3}+10 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{2} c d \,e^{2}+9 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a b c \,d^{2} e +10 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a \,c^{2} d^{3}}{10 c^{2} d} \] Input:
int((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^(1/2),x)
Output:
( - 6*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*e**2 - 20*sqrt(d + e*x)*sqrt(a - c*x**2)*a*c*d*e - 10*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c*d**2 - 4*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c*d*e*x - 9*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x* *2)/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a*b*c*e**3 - 20*int((sqrt(d + e *x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a*c**2*d *e**2 - 3*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d*x** 2 - c*e*x**3),x)*b*c**2*d**2*e + 3*int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a *d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**2*b*e**3 + 10*int((sqrt(d + e*x)*s qrt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**2*c*d*e**2 + 9* int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x )*a*b*c*d**2*e + 10*int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c* d*x**2 - c*e*x**3),x)*a*c**2*d**3)/(10*c**2*d)