Integrand size = 26, antiderivative size = 172 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}+\frac {21 c^2 d^{9/2} \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac {21 c^2 d^{9/2} \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\sqrt [4]{b^2-4 a c}} \] Output:
-1/2*d*(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^2-7*c*d^3*(2*c*d*x+b*d)^(3/2)/(2* c*x^2+2*b*x+2*a)+21*c^2*d^(9/2)*arctan((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1 /4)/d^(1/2))/(-4*a*c+b^2)^(1/4)-21*c^2*d^(9/2)*arctanh((2*c*d*x+b*d)^(1/2) /(-4*a*c+b^2)^(1/4)/d^(1/2))/(-4*a*c+b^2)^(1/4)
Result contains complex when optimal does not.
Time = 1.50 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.49 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx=\left (\frac {1}{2}+\frac {i}{2}\right ) c^2 (d (b+2 c x))^{9/2} \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (b^2+11 b c x+c \left (7 a+11 c x^2\right )\right )}{c^2 (b+2 c x)^3 (a+x (b+c x))^2}-\frac {21 \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)^{9/2}}+\frac {21 \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)^{9/2}}-\frac {21 \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)^{9/2}}\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^3,x]
Output:
(1/2 + I/2)*c^2*(d*(b + 2*c*x))^(9/2)*(((-1/2 + I/2)*(b^2 + 11*b*c*x + c*( 7*a + 11*c*x^2)))/(c^2*(b + 2*c*x)^3*(a + x*(b + c*x))^2) - (21*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/((b^2 - 4*a*c)^(1/4)*(b + 2*c*x)^(9/2)) + (21*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1 /4)])/((b^2 - 4*a*c)^(1/4)*(b + 2*c*x)^(9/2)) - (21*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 - 4*a*c)^(1/4)*(b + 2*c*x)^(9/2)))
Time = 0.35 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1110, 1110, 1118, 27, 25, 266, 827, 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.
| \(\displaystyle \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle \frac {7}{2} c d^2 \int \frac {(b d+2 c x d)^{5/2}}{\left (c x^2+b x+a\right )^2}dx-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle \frac {7}{2} c d^2 \left (3 c d^2 \int \frac {\sqrt {b d+2 c x d}}{c x^2+b x+a}dx-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1118 |
| \(\displaystyle \frac {7}{2} c d^2 \left (\frac {3}{2} d \int \frac {4 c d^2 \sqrt {b d+2 c x d}}{\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2}d(b d+2 c x d)-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{2} c d^2 \left (6 c d^3 \int -\frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7}{2} c d^2 \left (-6 c d^3 \int \frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {7}{2} c d^2 \left (-12 c d^3 \int \frac {b d+2 c x d}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {7}{2} c d^2 \left (-12 c d^3 \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {1}{2} \int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}\right )-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {7}{2} c d^2 \left (-12 c d^3 \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7}{2} c d^2 \left (-12 c d^3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {d (b d+2 c d x)^{3/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^3,x]
Output:
-1/2*(d*(b*d + 2*c*d*x)^(7/2))/(a + b*x + c*x^2)^2 + (7*c*d^2*(-((d*(b*d + 2*c*d*x)^(3/2))/(a + b*x + c*x^2)) - 12*c*d^3*(-1/2*ArcTan[Sqrt[b*d + 2*c *d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]/((b^2 - 4*a*c)^(1/4)*Sqrt[d]) + ArcTa nh[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]/(2*(b^2 - 4*a*c)^(1/ 4)*Sqrt[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[((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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) 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[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. \(304\) vs. \(2(148)=296\).
Time = 1.43 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.77
| method | result | size |
| pseudoelliptic | \(\frac {c^{2} d^{5} \left (-\frac {\left (d \left (2 c x +b \right )\right )^{\frac {3}{2}} \left (11 c^{2} x^{2}+11 c b x +7 a c +b^{2}\right )}{d^{2} c^{2} \left (c \,x^{2}+b x +a \right )^{2}}+\frac {21 \ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right ) \sqrt {2}}{2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+\frac {21 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right ) \sqrt {2}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+\frac {21 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right ) \sqrt {2}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )}{2}\) | \(305\) |
| derivativedivides | \(64 c^{2} d^{5} \left (\frac {-\frac {11 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (-\frac {7}{128} a \,d^{2} c +\frac {7}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(309\) |
| default | \(64 c^{2} d^{5} \left (\frac {-\frac {11 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (-\frac {7}{128} a \,d^{2} c +\frac {7}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(309\) |
Input:
int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/2*c^2*d^5*(-(d*(2*c*x+b))^(3/2)/d^2*(11*c^2*x^2+11*b*c*x+7*a*c+b^2)/c^2/ (c*x^2+b*x+a)^2+21/2*ln(((d^2*(4*a*c-b^2))^(1/2)-(d^2*(4*a*c-b^2))^(1/4)*( d*(2*c*x+b))^(1/2)*2^(1/2)+d*(2*c*x+b))/((d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x +b))^(1/2)*2^(1/2)+(d^2*(4*a*c-b^2))^(1/2)+d*(2*c*x+b)))/(d^2*(4*a*c-b^2)) ^(1/4)*2^(1/2)+21*arctan(2^(1/2)/(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b))^(1/ 2)+1)/(d^2*(4*a*c-b^2))^(1/4)*2^(1/2)+21*arctan(2^(1/2)/(d^2*(4*a*c-b^2))^ (1/4)*(d*(2*c*x+b))^(1/2)-1)/(d^2*(4*a*c-b^2))^(1/4)*2^(1/2))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 543, normalized size of antiderivative = 3.16 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {21 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (9261 \, \sqrt {2 \, c d x + b d} c^{6} d^{13} + 9261 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {3}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) - 21 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (9261 \, \sqrt {2 \, c d x + b d} c^{6} d^{13} - 9261 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {3}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) + 21 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {1}{4}} {\left (i \, c^{2} x^{4} + 2 i \, b c x^{3} + 2 i \, a b x + i \, {\left (b^{2} + 2 \, a c\right )} x^{2} + i \, a^{2}\right )} \log \left (9261 \, \sqrt {2 \, c d x + b d} c^{6} d^{13} - 9261 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {3}{4}} {\left (i \, b^{2} - 4 i \, a c\right )}\right ) + 21 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {1}{4}} {\left (-i \, c^{2} x^{4} - 2 i \, b c x^{3} - 2 i \, a b x - i \, {\left (b^{2} + 2 \, a c\right )} x^{2} - i \, a^{2}\right )} \log \left (9261 \, \sqrt {2 \, c d x + b d} c^{6} d^{13} - 9261 \, \left (\frac {c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac {3}{4}} {\left (-i \, b^{2} + 4 i \, a c\right )}\right ) + {\left (22 \, c^{3} d^{4} x^{3} + 33 \, b c^{2} d^{4} x^{2} + {\left (13 \, b^{2} c + 14 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 7 \, a b c\right )} d^{4}\right )} \sqrt {2 \, c d x + b d}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \] Input:
integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
-1/2*(21*(c^8*d^18/(b^2 - 4*a*c))^(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(9261*sqrt(2*c*d*x + b*d)*c^6*d^13 + 9261*(c^8* d^18/(b^2 - 4*a*c))^(3/4)*(b^2 - 4*a*c)) - 21*(c^8*d^18/(b^2 - 4*a*c))^(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(9261*sqrt (2*c*d*x + b*d)*c^6*d^13 - 9261*(c^8*d^18/(b^2 - 4*a*c))^(3/4)*(b^2 - 4*a* c)) + 21*(c^8*d^18/(b^2 - 4*a*c))^(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(9261*sqrt(2*c*d*x + b*d)*c^6*d^13 - 9261*(c^8*d^18/(b^2 - 4*a*c))^(3/4)*(I*b^2 - 4*I*a*c)) + 21*(c^8*d^18/(b^2 - 4*a*c))^(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(9261*sqrt(2*c*d*x + b*d)*c^6*d^13 - 9261*(c^8*d^18/(b^2 - 4*a*c))^(3/4)*(-I*b^2 + 4*I*a*c)) + (22*c^3*d^4*x^3 + 33*b*c^2*d^4*x^2 + ( 13*b^2*c + 14*a*c^2)*d^4*x + (b^3 + 7*a*b*c)*d^4)*sqrt(2*c*d*x + b*d))/(c^ 2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)
Timed out. \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^(9/2)/(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. 510 vs. \(2 (144) = 288\).
Time = 0.17 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.97 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} - \frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} + \frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} - \frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} d^{3} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} + \frac {2 \, {\left (7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{7} - 28 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{3} d^{7} - 11 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2} d^{5}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \] Input:
integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
-21*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^3*arctan(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))/(sqrt(2)*b^2 - 4*sqrt(2)*a*c) - 21*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d ^3*arctan(-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))/(sqrt(2)*b^2 - 4*sqrt(2)*a*c) + 21/2*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^3*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) + sqrt(-b^2*d^2 + 4*a*c*d^2)) /(sqrt(2)*b^2 - 4*sqrt(2)*a*c) - 21/2*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^3 *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) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^2 - 4*sqrt(2)*a*c) + 2*(7*(2* c*d*x + b*d)^(3/2)*b^2*c^2*d^7 - 28*(2*c*d*x + b*d)^(3/2)*a*c^3*d^7 - 11*( 2*c*d*x + b*d)^(7/2)*c^2*d^5)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2
Time = 5.23 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.25 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {21\,c^2\,d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )}{{\left (b^2-4\,a\,c\right )}^{1/4}}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (56\,a\,c^3\,d^7-14\,b^2\,c^2\,d^7\right )+22\,c^2\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}-\frac {21\,c^2\,d^{9/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )}{{\left (b^2-4\,a\,c\right )}^{1/4}} \] Input:
int((b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^3,x)
Output:
(21*c^2*d^(9/2)*atan((b*d + 2*c*d*x)^(1/2)/(d^(1/2)*(b^2 - 4*a*c)^(1/4)))) /(b^2 - 4*a*c)^(1/4) - ((b*d + 2*c*d*x)^(3/2)*(56*a*c^3*d^7 - 14*b^2*c^2*d ^7) + 22*c^2*d^5*(b*d + 2*c*d*x)^(7/2))/((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) - (21*c^2*d^(9/2)*atanh((b*d + 2*c*d*x)^(1/2)/(d^(1/2)*(b^2 - 4*a*c)^(1/4)) ))/(b^2 - 4*a*c)^(1/4)
Time = 0.24 (sec) , antiderivative size = 1815, normalized size of antiderivative = 10.55 \[ \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^3,x)
Output:
(sqrt(d)*d**4*( - 42*(4*a*c - b**2)**(3/4)*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* *2 - 84*(4*a*c - b**2)**(3/4)*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**2*x - 84*(4*a *c - b**2)**(3/4)*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*c**3*x**2 - 42*(4*a*c - b**2)* *(3/4)*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**2*c**2*x**2 - 84*(4*a*c - b**2)**(3/4)*s qrt(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*c**3*x**3 - 42*(4*a*c - b**2)**(3/4)*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)))*c**4*x**4 + 42*(4*a*c - b**2)**(3/4)*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**2 + 84*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sq rt(2) + 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*b*c**2*x + 8 4*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sq rt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*c**3*x**2 + 42*(4*a*c - b**2)**(3/4)*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**2*c**2*x**2 + 84*(4*a*c - b**2)**( 3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sqrt(b + 2*c*x))/(...