Integrand size = 26, antiderivative size = 224 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=234 c^2 \left (b^2-4 a c\right ) d^7 \sqrt {b d+2 c d x}+\frac {234}{5} c^2 d^5 (b d+2 c d x)^{5/2}-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}-117 c^2 \left (b^2-4 a c\right )^{5/4} d^{15/2} \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-117 c^2 \left (b^2-4 a c\right )^{5/4} d^{15/2} \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \] Output:
234*c^2*(-4*a*c+b^2)*d^7*(2*c*d*x+b*d)^(1/2)+234/5*c^2*d^5*(2*c*d*x+b*d)^( 5/2)-1/2*d*(2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a)^2-13*c*d^3*(2*c*d*x+b*d)^(9/ 2)/(2*c*x^2+2*b*x+2*a)-117*c^2*(-4*a*c+b^2)^(5/4)*d^(15/2)*arctan((2*c*d*x +b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))-117*c^2*(-4*a*c+b^2)^(5/4)*d^(15/2 )*arctanh((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))
Result contains complex when optimal does not.
Time = 2.22 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.58 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\left (\frac {1}{10}+\frac {i}{10}\right ) c^2 (d (b+2 c x))^{15/2} \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-585 b^6+7020 a b^4 c-28080 a^2 b^2 c^2+37440 a^3 c^3+1053 b^4 (b+2 c x)^2-8424 a b^2 c (b+2 c x)^2+16848 a^2 c^2 (b+2 c x)^2-416 b^2 (b+2 c x)^4+1664 a c (b+2 c x)^4-32 (b+2 c x)^6\right )}{c^2 (b+2 c x)^7 (a+x (b+c x))^2}-\frac {585 i \left (b^2-4 a c\right )^{5/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{5/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac {585 i \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{15/2}}\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x]
Output:
(1/10 + I/10)*c^2*(d*(b + 2*c*x))^(15/2)*(((-1/8 + I/8)*(-585*b^6 + 7020*a *b^4*c - 28080*a^2*b^2*c^2 + 37440*a^3*c^3 + 1053*b^4*(b + 2*c*x)^2 - 8424 *a*b^2*c*(b + 2*c*x)^2 + 16848*a^2*c^2*(b + 2*c*x)^2 - 416*b^2*(b + 2*c*x) ^4 + 1664*a*c*(b + 2*c*x)^4 - 32*(b + 2*c*x)^6))/(c^2*(b + 2*c*x)^7*(a + x *(b + c*x))^2) - ((585*I)*(b^2 - 4*a*c)^(5/4)*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(15/2) + ((585*I)*(b^2 - 4*a*c) ^(5/4)*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b + 2*c *x)^(15/2) + ((585*I)*(b^2 - 4*a*c)^(5/4)*ArcTanh[((1 + I)*(b^2 - 4*a*c)^( 1/4)*Sqrt[b + 2*c*x])/(Sqrt[b^2 - 4*a*c] + I*(b + 2*c*x))])/(b + 2*c*x)^(1 5/2))
Time = 0.45 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1110, 1110, 1116, 1116, 1118, 27, 25, 266, 756, 216, 219}
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.
| \(\displaystyle \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle \frac {13}{2} c d^2 \int \frac {(b d+2 c x d)^{11/2}}{\left (c x^2+b x+a\right )^2}dx-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \int \frac {(b d+2 c x d)^{7/2}}{c x^2+b x+a}dx-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \int \frac {(b d+2 c x d)^{3/2}}{c x^2+b x+a}dx+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \left (c x^2+b x+a\right )}dx+4 d \sqrt {b d+2 c d x}\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1118 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (\frac {d \left (b^2-4 a c\right ) \int \frac {4 c d^2}{\sqrt {b d+2 c x d} \left (\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2\right )}d(b d+2 c x d)}{2 c}+4 d \sqrt {b d+2 c d x}\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (2 d^3 \left (b^2-4 a c\right ) \int -\frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)+4 d \sqrt {b d+2 c d x}\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (4 d \sqrt {b d+2 c d x}-2 d^3 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \int \frac {1}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}\right )\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {13}{2} c d^2 \left (9 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )\right )+\frac {4}{5} d (b d+2 c d x)^{5/2}\right )-\frac {d (b d+2 c d x)^{9/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x]
Output:
-1/2*(d*(b*d + 2*c*d*x)^(13/2))/(a + b*x + c*x^2)^2 + (13*c*d^2*(-((d*(b*d + 2*c*d*x)^(9/2))/(a + b*x + c*x^2)) + 9*c*d^2*((4*d*(b*d + 2*c*d*x)^(5/2 ))/5 + (b^2 - 4*a*c)*d^2*(4*d*Sqrt[b*d + 2*c*d*x] - 4*(b^2 - 4*a*c)*d^3*(A rcTan[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]/(2*(b^2 - 4*a*c)^ (3/4)*d^(3/2)) + ArcTanh[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d]) ]/(2*(b^2 - 4*a*c)^(3/4)*d^(3/2)))))))/2
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d*e*((m - 1)/(b*(p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^ 2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && N eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[1/e Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(431\) vs. \(2(194)=388\).
Time = 1.60 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.93
| method | result | size |
| derivativedivides | \(64 c^{2} d^{5} \left (-12 a c \,d^{2} \sqrt {2 c d x +b d}+3 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+d^{4} \left (\frac {16 \left (-\frac {25}{32} a^{2} c^{2}+\frac {25}{64} c a \,b^{2}-\frac {25}{512} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {21}{8} a^{3} c^{3} d^{2}+\frac {63}{32} a^{2} b^{2} c^{2} d^{2}-\frac {63}{128} a \,b^{4} c \,d^{2}+\frac {21}{512} d^{2} b^{6}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {117 \left (a^{2} c^{2}-\frac {1}{2} c a \,b^{2}+\frac {1}{16} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(432\) |
| default | \(64 c^{2} d^{5} \left (-12 a c \,d^{2} \sqrt {2 c d x +b d}+3 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+d^{4} \left (\frac {16 \left (-\frac {25}{32} a^{2} c^{2}+\frac {25}{64} c a \,b^{2}-\frac {25}{512} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {21}{8} a^{3} c^{3} d^{2}+\frac {63}{32} a^{2} b^{2} c^{2} d^{2}-\frac {63}{128} a \,b^{4} c \,d^{2}+\frac {21}{512} d^{2} b^{6}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {117 \left (a^{2} c^{2}-\frac {1}{2} c a \,b^{2}+\frac {1}{16} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(432\) |
| pseudoelliptic | \(\frac {234 \left (-\frac {31 \left (-\frac {32 c^{4} x^{4}}{93}-\frac {64 x^{2} \left (b x +a \right ) c^{3}}{93}+\left (-\frac {32}{93} b^{2} x^{2}-\frac {64}{93} a b x +a^{2}\right ) c^{2}-\frac {125 c a \,b^{2}}{186}+\frac {125 b^{4}}{1488}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (d \left (2 c x +b \right )\right )^{\frac {5}{2}}}{195}+\frac {d^{2} \left (4 a c -b^{2}\right ) \left (-8 \left (\frac {32 c^{4} x^{4}}{39}+\frac {64 x^{2} \left (b x +a \right ) c^{3}}{39}+\left (\frac {32}{39} b^{2} x^{2}+\frac {64}{39} a b x +a^{2}\right ) c^{2}-\frac {7 c a \,b^{2}}{78}+\frac {7 b^{4}}{624}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}+c^{2} d^{2} \sqrt {2}\, \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2} \left (\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right )\right )\right )}{8}\right ) d^{5}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (c \,x^{2}+b x +a \right )^{2}}\) | \(435\) |
Input:
int((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
64*c^2*d^5*(-12*a*c*d^2*(2*c*d*x+b*d)^(1/2)+3*b^2*d^2*(2*c*d*x+b*d)^(1/2)+ 1/5*(2*c*d*x+b*d)^(5/2)+d^4*(16*((-25/32*a^2*c^2+25/64*c*a*b^2-25/512*b^4) *(2*c*d*x+b*d)^(5/2)+(-21/8*a^3*c^3*d^2+63/32*a^2*b^2*c^2*d^2-63/128*a*b^4 *c*d^2+21/512*d^2*b^6)*(2*c*d*x+b*d)^(1/2))/((2*c*d*x+b*d)^2+4*a*d^2*c-b^2 *d^2)^2+117/16*(a^2*c^2-1/2*c*a*b^2+1/16*b^4)/(4*a*c*d^2-b^2*d^2)^(3/4)*2^ (1/2)*(ln((2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/ 2)+(4*a*c*d^2-b^2*d^2)^(1/2))/(2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c* d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2)))+2*arctan(2^(1/2)/(4*a*c *d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-2*arctan(-2^(1/2)/(4*a*c*d^2-b^ 2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1))))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 1029, normalized size of antiderivative = 4.59 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
1/10*(585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2* a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*log(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d^7 + 117*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b ^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)) - 585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)*(-I*c^2*x^4 - 2*I*b*c*x^3 - 2*I*a*b*x - I*(b^ 2 + 2*a*c)*x^2 - I*a^2)*log(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d ^7 + 117*I*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)) - 585*((b^10*c^8 - 20*a *b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024* a^5*c^13)*d^30)^(1/4)*(I*c^2*x^4 + 2*I*b*c*x^3 + 2*I*a*b*x + I*(b^2 + 2*a* c)*x^2 + I*a^2)*log(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d^7 - 117 *I*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280* a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)) - 585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13 )*d^30)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*lo g(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d^7 - 117*((b^10*c^8 - 20*a *b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024* a^5*c^13)*d^30)^(1/4)) + (512*c^6*d^7*x^6 + 1536*b*c^5*d^7*x^5 + 512*(7...
Timed out. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (190) = 380\).
Time = 0.19 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.36 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
192*sqrt(2*c*d*x + b*d)*b^2*c^2*d^7 - 768*sqrt(2*c*d*x + b*d)*a*c^3*d^7 + 64/5*(2*c*d*x + b*d)^(5/2)*c^2*d^5 - 117/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2) ^(1/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d^7)*arc tan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b *d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4)) - 117/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2) ^(1/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d^7)*arc tan(-1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4)) - 117/4*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2 )^(1/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d^7)*lo g(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2)) + 117/4*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/ 4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*a*c^3*d^7)*log(2*c *d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sq rt(-b^2*d^2 + 4*a*c*d^2)) + 2*(21*sqrt(2*c*d*x + b*d)*b^6*c^2*d^11 - 252*s qrt(2*c*d*x + b*d)*a*b^4*c^3*d^11 + 1008*sqrt(2*c*d*x + b*d)*a^2*b^2*c^4*d ^11 - 1344*sqrt(2*c*d*x + b*d)*a^3*c^5*d^11 - 25*(2*c*d*x + b*d)^(5/2)*b^4 *c^2*d^9 + 200*(2*c*d*x + b*d)^(5/2)*a*b^2*c^3*d^9 - 400*(2*c*d*x + b*d)^( 5/2)*a^2*c^4*d^9)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2
Time = 5.24 (sec) , antiderivative size = 966, normalized size of antiderivative = 4.31 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x)
Output:
(64*c^2*d^5*(b*d + 2*c*d*x)^(5/2))/5 - ((b*d + 2*c*d*x)^(5/2)*(800*a^2*c^4 *d^9 + 50*b^4*c^2*d^9 - 400*a*b^2*c^3*d^9) + (b*d + 2*c*d*x)^(1/2)*(2688*a ^3*c^5*d^11 - 42*b^6*c^2*d^11 + 504*a*b^4*c^3*d^11 - 2016*a^2*b^2*c^4*d^11 ))/((b*d + 2*c*d*x)^4 - (b*d + 2*c*d*x)^2*(2*b^2*d^2 - 8*a*c*d^2) + b^4*d^ 4 + 16*a^2*c^2*d^4 - 8*a*b^2*c*d^4) - 192*c^2*d^7*(b*d + 2*c*d*x)^(1/2)*(4 *a*c - b^2) + c^2*d^(15/2)*atan(((c^2*d^(15/2)*((b*d + 2*c*d*x)^(1/2)*(560 70144*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - 3504384*a*b^6*c^5*d^18 + 210263 04*a^2*b^4*c^6*d^18 - 56070144*a^3*b^2*c^7*d^18) - (117*c^2*d^(15/2)*(b^2 - 4*a*c)^(5/4)*(239616*a^3*c^5*d^11 - 3744*b^6*c^2*d^11 + 44928*a*b^4*c^3* d^11 - 179712*a^2*b^2*c^4*d^11))/2)*(b^2 - 4*a*c)^(5/4)*117i)/2 + (c^2*d^( 15/2)*((b*d + 2*c*d*x)^(1/2)*(56070144*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - 3504384*a*b^6*c^5*d^18 + 21026304*a^2*b^4*c^6*d^18 - 56070144*a^3*b^2*c^ 7*d^18) + (117*c^2*d^(15/2)*(b^2 - 4*a*c)^(5/4)*(239616*a^3*c^5*d^11 - 374 4*b^6*c^2*d^11 + 44928*a*b^4*c^3*d^11 - 179712*a^2*b^2*c^4*d^11))/2)*(b^2 - 4*a*c)^(5/4)*117i)/2)/((117*c^2*d^(15/2)*((b*d + 2*c*d*x)^(1/2)*(5607014 4*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - 3504384*a*b^6*c^5*d^18 + 21026304*a ^2*b^4*c^6*d^18 - 56070144*a^3*b^2*c^7*d^18) - (117*c^2*d^(15/2)*(b^2 - 4* a*c)^(5/4)*(239616*a^3*c^5*d^11 - 3744*b^6*c^2*d^11 + 44928*a*b^4*c^3*d^11 - 179712*a^2*b^2*c^4*d^11))/2)*(b^2 - 4*a*c)^(5/4))/2 - (117*c^2*d^(15/2) *((b*d + 2*c*d*x)^(1/2)*(56070144*a^4*c^8*d^18 + 219024*b^8*c^4*d^18 - ...
Time = 0.22 (sec) , antiderivative size = 3221, normalized size of antiderivative = 14.38 \[ \int \frac {(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^3,x)
Output:
(sqrt(d)*d**7*( - 4680*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c - b**2)* *(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a**3* c**3 + 1170*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt (2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a**2*b**2*c**2 - 9360*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a**2*b*c**3*x - 9360*( 4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt( b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a**2*c**4*x**2 + 2340*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2* c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*b**3*c**2*x - 2340*(4*a*c - b**2) **(1/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/( (4*a*c - b**2)**(1/4)*sqrt(2)))*a*b**2*c**3*x**2 - 9360*(4*a*c - b**2)**(1 /4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a *c - b**2)**(1/4)*sqrt(2)))*a*b*c**4*x**3 - 4680*(4*a*c - b**2)**(1/4)*sqr t(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b* *2)**(1/4)*sqrt(2)))*a*c**5*x**4 + 1170*(4*a*c - b**2)**(1/4)*sqrt(2)*atan (((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4 )*sqrt(2)))*b**4*c**2*x**2 + 2340*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a *c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt (2)))*b**3*c**3*x**3 + 1170*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c ...