Integrand size = 41, antiderivative size = 274 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {c^2-d^2 x^2}}{2 d^4 (c+d x)^{5/2}}+\frac {\left (17 c^2 C d-9 B c d^2+A d^3-25 c^3 D\right ) \sqrt {c^2-d^2 x^2}}{8 c d^4 (c+d x)^{3/2}}+\frac {2 (3 C d-11 c D) \sqrt {c^2-d^2 x^2}}{3 d^4 \sqrt {c+d x}}+\frac {2 D \sqrt {c+d x} \sqrt {c^2-d^2 x^2}}{3 d^4}-\frac {\left (47 c^2 C d-7 B c d^2-A d^3-119 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{8 \sqrt {2} c^{3/2} d^4} \] Output:
-1/2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^(5/2)+ 1/8*(A*d^3-9*B*c*d^2+17*C*c^2*d-25*D*c^3)*(-d^2*x^2+c^2)^(1/2)/c/d^4/(d*x+ c)^(3/2)+2/3*(3*C*d-11*D*c)*(-d^2*x^2+c^2)^(1/2)/d^4/(d*x+c)^(1/2)+2/3*D*( d*x+c)^(1/2)*(-d^2*x^2+c^2)^(1/2)/d^4-1/16*(-A*d^3-7*B*c*d^2+47*C*c^2*d-11 9*D*c^3)*arctanh(2^(1/2)*c^(1/2)*(d*x+c)^(1/2)/(-d^2*x^2+c^2)^(1/2))*2^(1/ 2)/c^(3/2)/d^4
Time = 1.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {c^2-d^2 x^2} \left (223 c^4 D-3 A d^4 x+c^3 (-87 C d+379 d D x)+c^2 d^2 (15 B+x (-147 C+128 D x))+c d^3 (9 A+x (27 B-16 x (3 C+D x)))\right )}{(c+d x)^{5/2}}+3 \sqrt {2} \left (-47 c^2 C d+7 B c d^2+A d^3+119 c^3 D\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{48 c^{3/2} d^4} \] Input:
Integrate[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2), x]
Output:
((-2*Sqrt[c]*Sqrt[c^2 - d^2*x^2]*(223*c^4*D - 3*A*d^4*x + c^3*(-87*C*d + 3 79*d*D*x) + c^2*d^2*(15*B + x*(-147*C + 128*D*x)) + c*d^3*(9*A + x*(27*B - 16*x*(3*C + D*x)))))/(c + d*x)^(5/2) + 3*Sqrt[2]*(-47*c^2*C*d + 7*B*c*d^2 + A*d^3 + 119*c^3*D)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])/Sqrt[c^2 - d ^2*x^2]])/(48*c^(3/2)*d^4)
Time = 1.18 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {2170, 27, 2170, 27, 671, 465, 471, 221}
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.
| \(\displaystyle \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2170 |
| \(\displaystyle -\frac {2 \int -\frac {3 \sqrt {c^2-d^2 x^2} \left ((C d-3 c D) x^2 d^4+\left (B d^2-3 c^2 D\right ) x d^3+\left (A d^3-c^3 D\right ) d^2\right )}{2 (c+d x)^{7/2}}dx}{3 d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c^2-d^2 x^2} \left ((C d-3 c D) x^2 d^4+\left (B d^2-3 c^2 D\right ) x d^3+\left (A d^3-c^3 D\right ) d^2\right )}{(c+d x)^{7/2}}dx}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 2170 |
| \(\displaystyle \frac {-\frac {2 \int \frac {d^6 \left (-14 D c^3+5 C d c^2-A d^3+d \left (-15 D c^2+6 C d c-B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{2 (c+d x)^{7/2}}dx}{d^4}-\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-3 c D)}{(c+d x)^{5/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (-\int \frac {\left (-14 D c^3+5 C d c^2-A d^3+d \left (-15 D c^2+6 C d c-B d^2\right ) x\right ) \sqrt {c^2-d^2 x^2}}{(c+d x)^{7/2}}dx\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-3 c D)}{(c+d x)^{5/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 671 |
| \(\displaystyle \frac {-\left (d^2 \left (\frac {\left (-A d^3-7 B c d^2-119 c^3 D+47 c^2 C d\right ) \int \frac {\sqrt {c^2-d^2 x^2}}{(c+d x)^{5/2}}dx}{8 c}+\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{7/2}}\right )\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-3 c D)}{(c+d x)^{5/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 465 |
| \(\displaystyle \frac {-\left (d^2 \left (\frac {\left (-A d^3-7 B c d^2-119 c^3 D+47 c^2 C d\right ) \left (-\frac {1}{2} \int \frac {1}{\sqrt {c+d x} \sqrt {c^2-d^2 x^2}}dx-\frac {\sqrt {c^2-d^2 x^2}}{d (c+d x)^{3/2}}\right )}{8 c}+\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{7/2}}\right )\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-3 c D)}{(c+d x)^{5/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 471 |
| \(\displaystyle \frac {-\left (d^2 \left (\frac {\left (-A d^3-7 B c d^2-119 c^3 D+47 c^2 C d\right ) \left (-d \int \frac {1}{\frac {d^2 \left (c^2-d^2 x^2\right )}{c+d x}-2 c d^2}d\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {c+d x}}-\frac {\sqrt {c^2-d^2 x^2}}{d (c+d x)^{3/2}}\right )}{8 c}+\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{7/2}}\right )\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-3 c D)}{(c+d x)^{5/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\left (d^2 \left (\frac {\left (\frac {\text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {2} \sqrt {c} d}-\frac {\sqrt {c^2-d^2 x^2}}{d (c+d x)^{3/2}}\right ) \left (-A d^3-7 B c d^2-119 c^3 D+47 c^2 C d\right )}{8 c}+\frac {\left (c^2-d^2 x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 c d (c+d x)^{7/2}}\right )\right )-\frac {2 d \left (c^2-d^2 x^2\right )^{3/2} (C d-3 c D)}{(c+d x)^{5/2}}}{d^5}-\frac {2 D \left (c^2-d^2 x^2\right )^{3/2}}{3 d^4 (c+d x)^{3/2}}\) |
Input:
Int[(Sqrt[c^2 - d^2*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(7/2),x]
Output:
(-2*D*(c^2 - d^2*x^2)^(3/2))/(3*d^4*(c + d*x)^(3/2)) + ((-2*d*(C*d - 3*c*D )*(c^2 - d^2*x^2)^(3/2))/(c + d*x)^(5/2) - d^2*(((c^2*C*d - B*c*d^2 + A*d^ 3 - c^3*D)*(c^2 - d^2*x^2)^(3/2))/(4*c*d*(c + d*x)^(7/2)) + ((47*c^2*C*d - 7*B*c*d^2 - A*d^3 - 119*c^3*D)*(-(Sqrt[c^2 - d^2*x^2]/(d*(c + d*x)^(3/2)) ) + ArcTanh[Sqrt[c^2 - d^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[c + d*x])]/(Sqrt[2]* Sqrt[c]*d)))/(8*c)))/d^5
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + p + 1))), x] - Simp[b*(p/(d^2*(n + p + 1))) Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LtQ[n, -2] || EqQ[n + 2*p + 1, 0]) && NeQ[n + p + 1, 0] && IntegerQ[2*p]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[2*d Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] ], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - b*d*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(581\) vs. \(2(236)=472\).
Time = 0.36 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.12
| method | result | size |
| default | \(\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) d^{5} x^{2}+21 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{4} x^{2}-141 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3} x^{2}+357 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2} x^{2}+32 D c^{\frac {3}{2}} d^{3} x^{3} \sqrt {-d x +c}+6 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \,d^{4} x +42 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3} x -282 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2} x +96 C \sqrt {-d x +c}\, c^{\frac {3}{2}} d^{3} x^{2}+714 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d x -256 D \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2} x^{2}+3 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2} d^{3}+6 A \,d^{4} x \sqrt {-d x +c}\, \sqrt {c}+21 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} d^{2}-54 B \,c^{\frac {3}{2}} d^{3} x \sqrt {-d x +c}-141 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} d +294 C \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2} x +357 D \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-d x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{5}-758 D \sqrt {-d x +c}\, c^{\frac {7}{2}} d x -18 A \sqrt {-d x +c}\, c^{\frac {3}{2}} d^{3}-30 B \sqrt {-d x +c}\, c^{\frac {5}{2}} d^{2}+174 C \sqrt {-d x +c}\, c^{\frac {7}{2}} d -446 D \sqrt {-d x +c}\, c^{\frac {9}{2}}\right )}{48 c^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{2}} \sqrt {-d x +c}\, d^{4}}\) | \(582\) |
Input:
int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x,method=_RETUR NVERBOSE)
Output:
1/48*(-d^2*x^2+c^2)^(1/2)/c^(3/2)*(3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)* 2^(1/2)/c^(1/2))*d^5*x^2+21*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c ^(1/2))*c*d^4*x^2-141*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2) )*c^2*d^3*x^2+357*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^ 3*d^2*x^2+32*D*c^(3/2)*d^3*x^3*(-d*x+c)^(1/2)+6*A*2^(1/2)*arctanh(1/2*(-d* x+c)^(1/2)*2^(1/2)/c^(1/2))*c*d^4*x+42*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2 )*2^(1/2)/c^(1/2))*c^2*d^3*x-282*C*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1 /2)/c^(1/2))*c^3*d^2*x+96*C*(-d*x+c)^(1/2)*c^(3/2)*d^3*x^2+714*D*2^(1/2)*a rctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d*x-256*D*(-d*x+c)^(1/2)*c^ (5/2)*d^2*x^2+3*A*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^2* d^3+6*A*d^4*x*(-d*x+c)^(1/2)*c^(1/2)+21*B*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/ 2)*2^(1/2)/c^(1/2))*c^3*d^2-54*B*c^(3/2)*d^3*x*(-d*x+c)^(1/2)-141*C*2^(1/2 )*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^4*d+294*C*(-d*x+c)^(1/2)*c ^(5/2)*d^2*x+357*D*2^(1/2)*arctanh(1/2*(-d*x+c)^(1/2)*2^(1/2)/c^(1/2))*c^5 -758*D*(-d*x+c)^(1/2)*c^(7/2)*d*x-18*A*(-d*x+c)^(1/2)*c^(3/2)*d^3-30*B*(-d *x+c)^(1/2)*c^(5/2)*d^2+174*C*(-d*x+c)^(1/2)*c^(7/2)*d-446*D*(-d*x+c)^(1/2 )*c^(9/2))/(d*x+c)^(5/2)/(-d*x+c)^(1/2)/d^4
Time = 0.11 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (119 \, D c^{6} - 47 \, C c^{5} d + 7 \, B c^{4} d^{2} + A c^{3} d^{3} + {\left (119 \, D c^{3} d^{3} - 47 \, C c^{2} d^{4} + 7 \, B c d^{5} + A d^{6}\right )} x^{3} + 3 \, {\left (119 \, D c^{4} d^{2} - 47 \, C c^{3} d^{3} + 7 \, B c^{2} d^{4} + A c d^{5}\right )} x^{2} + 3 \, {\left (119 \, D c^{5} d - 47 \, C c^{4} d^{2} + 7 \, B c^{3} d^{3} + A c^{2} d^{4}\right )} x\right )} \sqrt {c} \log \left (-\frac {d^{2} x^{2} - 2 \, c d x - 2 \, \sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {c} - 3 \, c^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 4 \, {\left (16 \, D c^{2} d^{3} x^{3} - 223 \, D c^{5} + 87 \, C c^{4} d - 15 \, B c^{3} d^{2} - 9 \, A c^{2} d^{3} - 16 \, {\left (8 \, D c^{3} d^{2} - 3 \, C c^{2} d^{3}\right )} x^{2} - {\left (379 \, D c^{4} d - 147 \, C c^{3} d^{2} + 27 \, B c^{2} d^{3} - 3 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{96 \, {\left (c^{2} d^{7} x^{3} + 3 \, c^{3} d^{6} x^{2} + 3 \, c^{4} d^{5} x + c^{5} d^{4}\right )}}, -\frac {3 \, \sqrt {2} {\left (119 \, D c^{6} - 47 \, C c^{5} d + 7 \, B c^{4} d^{2} + A c^{3} d^{3} + {\left (119 \, D c^{3} d^{3} - 47 \, C c^{2} d^{4} + 7 \, B c d^{5} + A d^{6}\right )} x^{3} + 3 \, {\left (119 \, D c^{4} d^{2} - 47 \, C c^{3} d^{3} + 7 \, B c^{2} d^{4} + A c d^{5}\right )} x^{2} + 3 \, {\left (119 \, D c^{5} d - 47 \, C c^{4} d^{2} + 7 \, B c^{3} d^{3} + A c^{2} d^{4}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c} \sqrt {-c}}{2 \, {\left (c d x + c^{2}\right )}}\right ) - 2 \, {\left (16 \, D c^{2} d^{3} x^{3} - 223 \, D c^{5} + 87 \, C c^{4} d - 15 \, B c^{3} d^{2} - 9 \, A c^{2} d^{3} - 16 \, {\left (8 \, D c^{3} d^{2} - 3 \, C c^{2} d^{3}\right )} x^{2} - {\left (379 \, D c^{4} d - 147 \, C c^{3} d^{2} + 27 \, B c^{2} d^{3} - 3 \, A c d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}} \sqrt {d x + c}}{48 \, {\left (c^{2} d^{7} x^{3} + 3 \, c^{3} d^{6} x^{2} + 3 \, c^{4} d^{5} x + c^{5} d^{4}\right )}}\right ] \] Input:
integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algori thm="fricas")
Output:
[1/96*(3*sqrt(2)*(119*D*c^6 - 47*C*c^5*d + 7*B*c^4*d^2 + A*c^3*d^3 + (119* D*c^3*d^3 - 47*C*c^2*d^4 + 7*B*c*d^5 + A*d^6)*x^3 + 3*(119*D*c^4*d^2 - 47* C*c^3*d^3 + 7*B*c^2*d^4 + A*c*d^5)*x^2 + 3*(119*D*c^5*d - 47*C*c^4*d^2 + 7 *B*c^3*d^3 + A*c^2*d^4)*x)*sqrt(c)*log(-(d^2*x^2 - 2*c*d*x - 2*sqrt(2)*sqr t(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(c) - 3*c^2)/(d^2*x^2 + 2*c*d*x + c^2) ) + 4*(16*D*c^2*d^3*x^3 - 223*D*c^5 + 87*C*c^4*d - 15*B*c^3*d^2 - 9*A*c^2* d^3 - 16*(8*D*c^3*d^2 - 3*C*c^2*d^3)*x^2 - (379*D*c^4*d - 147*C*c^3*d^2 + 27*B*c^2*d^3 - 3*A*c*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c))/(c^2*d^7* x^3 + 3*c^3*d^6*x^2 + 3*c^4*d^5*x + c^5*d^4), -1/48*(3*sqrt(2)*(119*D*c^6 - 47*C*c^5*d + 7*B*c^4*d^2 + A*c^3*d^3 + (119*D*c^3*d^3 - 47*C*c^2*d^4 + 7 *B*c*d^5 + A*d^6)*x^3 + 3*(119*D*c^4*d^2 - 47*C*c^3*d^3 + 7*B*c^2*d^4 + A* c*d^5)*x^2 + 3*(119*D*c^5*d - 47*C*c^4*d^2 + 7*B*c^3*d^3 + A*c^2*d^4)*x)*s qrt(-c)*arctan(1/2*sqrt(2)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)*sqrt(-c)/(c* d*x + c^2)) - 2*(16*D*c^2*d^3*x^3 - 223*D*c^5 + 87*C*c^4*d - 15*B*c^3*d^2 - 9*A*c^2*d^3 - 16*(8*D*c^3*d^2 - 3*C*c^2*d^3)*x^2 - (379*D*c^4*d - 147*C* c^3*d^2 + 27*B*c^2*d^3 - 3*A*c*d^4)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(d*x + c)) /(c^2*d^7*x^3 + 3*c^3*d^6*x^2 + 3*c^4*d^5*x + c^5*d^4)]
\[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\sqrt {- \left (- c + d x\right ) \left (c + d x\right )} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((-d**2*x**2+c**2)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(7/2),x)
Output:
Integral(sqrt(-(-c + d*x)*(c + d*x))*(A + B*x + C*x**2 + D*x**3)/(c + d*x) **(7/2), x)
\[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int { \frac {\sqrt {-d^{2} x^{2} + c^{2}} {\left (D x^{3} + C x^{2} + B x + A\right )}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algori thm="maxima")
Output:
integrate(sqrt(-d^2*x^2 + c^2)*(D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(7/2), x)
Time = 0.13 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=-\frac {32 \, {\left (-d x + c\right )}^{\frac {3}{2}} D + 288 \, \sqrt {-d x + c} D c - 96 \, \sqrt {-d x + c} C d + \frac {3 \, \sqrt {2} {\left (119 \, D c^{3} - 47 \, C c^{2} d + 7 \, B c d^{2} + A d^{3}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-d x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {6 \, {\left (25 \, {\left (-d x + c\right )}^{\frac {3}{2}} D c^{3} - 46 \, \sqrt {-d x + c} D c^{4} - 17 \, {\left (-d x + c\right )}^{\frac {3}{2}} C c^{2} d + 30 \, \sqrt {-d x + c} C c^{3} d + 9 \, {\left (-d x + c\right )}^{\frac {3}{2}} B c d^{2} - 14 \, \sqrt {-d x + c} B c^{2} d^{2} - {\left (-d x + c\right )}^{\frac {3}{2}} A d^{3} - 2 \, \sqrt {-d x + c} A c d^{3}\right )}}{{\left (d x + c\right )}^{2} c}}{48 \, d^{4}} \] Input:
integrate((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x, algori thm="giac")
Output:
-1/48*(32*(-d*x + c)^(3/2)*D + 288*sqrt(-d*x + c)*D*c - 96*sqrt(-d*x + c)* C*d + 3*sqrt(2)*(119*D*c^3 - 47*C*c^2*d + 7*B*c*d^2 + A*d^3)*arctan(1/2*sq rt(2)*sqrt(-d*x + c)/sqrt(-c))/(sqrt(-c)*c) - 6*(25*(-d*x + c)^(3/2)*D*c^3 - 46*sqrt(-d*x + c)*D*c^4 - 17*(-d*x + c)^(3/2)*C*c^2*d + 30*sqrt(-d*x + c)*C*c^3*d + 9*(-d*x + c)^(3/2)*B*c*d^2 - 14*sqrt(-d*x + c)*B*c^2*d^2 - (- d*x + c)^(3/2)*A*d^3 - 2*sqrt(-d*x + c)*A*c*d^3)/((d*x + c)^2*c))/d^4
Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{7/2}} \,d x \] Input:
int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2),x)
Output:
int(((c^2 - d^2*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(7/2), x)
Time = 0.84 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {c^2-d^2 x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{7/2}} \, dx =\text {Too large to display} \] Input:
int((-d^2*x^2+c^2)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(7/2),x)
Output:
( - 144*sqrt(c - d*x)*a*c**2*d**2 + 48*sqrt(c - d*x)*a*c*d**3*x - 240*sqrt (c - d*x)*b*c**3*d - 432*sqrt(c - d*x)*b*c**2*d**2*x - 2176*sqrt(c - d*x)* c**5 - 3712*sqrt(c - d*x)*c**4*d*x - 1280*sqrt(c - d*x)*c**3*d**2*x**2 + 2 56*sqrt(c - d*x)*c**2*d**3*x**3 - 24*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a*c**2*d**2 - 48*sqrt(c)*sqrt(2)*log(tan(asin (sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a*c*d**3*x - 24*sqrt(c)*sqrt(2)*log( tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*a*d**4*x**2 - 168*sqrt(c)*sq rt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**3*d - 336*sqr t(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2))*b*c**2*d**2 *x - 168*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)*sqrt(2)))/2)) *b*c*d**3*x**2 - 1728*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt(c)* sqrt(2)))/2))*c**5 - 3456*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d*x)/(sqrt (c)*sqrt(2)))/2))*c**4*d*x - 1728*sqrt(c)*sqrt(2)*log(tan(asin(sqrt(c + d* x)/(sqrt(c)*sqrt(2)))/2))*c**3*d**2*x**2 + 9*sqrt(c)*sqrt(2)*a*c**2*d**2 + 18*sqrt(c)*sqrt(2)*a*c*d**3*x + 9*sqrt(c)*sqrt(2)*a*d**4*x**2 + 135*sqrt( c)*sqrt(2)*b*c**3*d + 270*sqrt(c)*sqrt(2)*b*c**2*d**2*x + 135*sqrt(c)*sqrt (2)*b*c*d**3*x**2 + 2192*sqrt(c)*sqrt(2)*c**5 + 4384*sqrt(c)*sqrt(2)*c**4* d*x + 2192*sqrt(c)*sqrt(2)*c**3*d**2*x**2)/(384*c**2*d**3*(c**2 + 2*c*d*x + d**2*x**2))