Integrand size = 26, antiderivative size = 224 \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx=110 c^2 \left (b^2-4 a c\right ) d^7 (b d+2 c d x)^{3/2}+\frac {330}{7} c^2 d^5 (b d+2 c d x)^{7/2}-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {15 c d^3 (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )}+165 c^2 \left (b^2-4 a c\right )^{7/4} d^{17/2} \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-165 c^2 \left (b^2-4 a c\right )^{7/4} d^{17/2} \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \] Output:
110*c^2*(-4*a*c+b^2)*d^7*(2*c*d*x+b*d)^(3/2)+330/7*c^2*d^5*(2*c*d*x+b*d)^( 7/2)-1/2*d*(2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^2-15*c*d^3*(2*c*d*x+b*d)^(11 /2)/(2*c*x^2+2*b*x+2*a)+165*c^2*(-4*a*c+b^2)^(7/4)*d^(17/2)*arctan((2*c*d* x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))-165*c^2*(-4*a*c+b^2)^(7/4)*d^(17/ 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 = 1.96 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.53 \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {1}{14} (d (b+2 c x))^{17/2} \left (\frac {-7 b^6-189 b^5 c x+40 b^3 c^2 x \left (89 a+64 c x^2\right )+5 b^4 c \left (-21 a+167 c x^2\right )+40 b^2 c^2 \left (55 a^2+25 a c x^2+64 c^2 x^4\right )+16 b c^3 x \left (-605 a^2-320 a c x^2+96 c^2 x^4\right )-16 c^3 \left (385 a^3+605 a^2 c x^2+160 a c^2 x^4-32 c^3 x^6\right )}{(b+2 c x)^7 (a+x (b+c x))^2}+\frac {(1155+1155 i) c^2 \left (b^2-4 a c\right )^{7/4} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (b+i \sqrt {b^2-4 a c}+2 c x\right )}{\sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}\right )}{(b+2 c x)^{17/2}}-\frac {(1155+1155 i) c^2 \left (b^2-4 a c\right )^{7/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{i b+\sqrt {b^2-4 a c}+2 i c x}\right )}{(b+2 c x)^{17/2}}\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^(17/2)/(a + b*x + c*x^2)^3,x]
Output:
((d*(b + 2*c*x))^(17/2)*((-7*b^6 - 189*b^5*c*x + 40*b^3*c^2*x*(89*a + 64*c *x^2) + 5*b^4*c*(-21*a + 167*c*x^2) + 40*b^2*c^2*(55*a^2 + 25*a*c*x^2 + 64 *c^2*x^4) + 16*b*c^3*x*(-605*a^2 - 320*a*c*x^2 + 96*c^2*x^4) - 16*c^3*(385 *a^3 + 605*a^2*c*x^2 + 160*a*c^2*x^4 - 32*c^3*x^6))/((b + 2*c*x)^7*(a + x* (b + c*x))^2) + ((1155 + 1155*I)*c^2*(b^2 - 4*a*c)^(7/4)*ArcTan[((1/2 + I/ 2)*(b + I*Sqrt[b^2 - 4*a*c] + 2*c*x))/((b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x] )])/(b + 2*c*x)^(17/2) - ((1155 + 1155*I)*c^2*(b^2 - 4*a*c)^(7/4)*ArcTanh[ ((1 + I)*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(I*b + Sqrt[b^2 - 4*a*c] + ( 2*I)*c*x)])/(b + 2*c*x)^(17/2)))/14
Time = 0.45 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.09, 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, 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.
| \(\displaystyle \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle \frac {15}{2} c d^2 \int \frac {(b d+2 c x d)^{13/2}}{\left (c x^2+b x+a\right )^2}dx-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \int \frac {(b d+2 c x d)^{9/2}}{c x^2+b x+a}dx-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \int \frac {(b d+2 c x d)^{5/2}}{c x^2+b x+a}dx+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (d^2 \left (b^2-4 a c\right ) \int \frac {\sqrt {b d+2 c x d}}{c x^2+b x+a}dx+\frac {4}{3} d (b d+2 c d x)^{3/2}\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1118 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (\frac {d \left (b^2-4 a c\right ) \int \frac {4 c d^2 \sqrt {b d+2 c x d}}{\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2}d(b d+2 c x d)}{2 c}+\frac {4}{3} d (b d+2 c d x)^{3/2}\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (2 d^3 \left (b^2-4 a c\right ) \int -\frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)+\frac {4}{3} d (b d+2 c d x)^{3/2}\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (\frac {4}{3} d (b d+2 c d x)^{3/2}-2 d^3 \left (b^2-4 a c\right ) \int \frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (\frac {4}{3} d (b d+2 c d x)^{3/2}-4 d^3 \left (b^2-4 a c\right ) \int \frac {b d+2 c x d}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (\frac {4}{3} d (b d+2 c d x)^{3/2}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {1}{2} \int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}\right )\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (\frac {4}{3} d (b d+2 c d x)^{3/2}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {15}{2} c d^2 \left (11 c d^2 \left (d^2 \left (b^2-4 a c\right ) \left (\frac {4}{3} d (b d+2 c d x)^{3/2}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\right )-\frac {d (b d+2 c d x)^{11/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(b*d + 2*c*d*x)^(17/2)/(a + b*x + c*x^2)^3,x]
Output:
-1/2*(d*(b*d + 2*c*d*x)^(15/2))/(a + b*x + c*x^2)^2 + (15*c*d^2*(-((d*(b*d + 2*c*d*x)^(11/2))/(a + b*x + c*x^2)) + 11*c*d^2*((4*d*(b*d + 2*c*d*x)^(7 /2))/7 + (b^2 - 4*a*c)*d^2*((4*d*(b*d + 2*c*d*x)^(3/2))/3 - 4*(b^2 - 4*a*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]) + ArcTanh[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[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 = 1.65 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.93
| method | result | size |
| derivativedivides | \(64 c^{2} d^{5} \left (-4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}+b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+d^{4} \left (\frac {16 \left (-\frac {27}{32} a^{2} c^{2}+\frac {27}{64} c a \,b^{2}-\frac {27}{512} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {7}{2}}+16 \left (-\frac {23}{8} a^{3} c^{3} d^{2}+\frac {69}{32} a^{2} b^{2} c^{2} d^{2}-\frac {69}{128} a \,b^{4} c \,d^{2}+\frac {23}{512} d^{2} b^{6}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {\left (\frac {165}{2} a^{2} c^{2}-\frac {165}{4} c a \,b^{2}+\frac {165}{32} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\right )\) | \(432\) |
| default | \(64 c^{2} d^{5} \left (-4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}+b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+d^{4} \left (\frac {16 \left (-\frac {27}{32} a^{2} c^{2}+\frac {27}{64} c a \,b^{2}-\frac {27}{512} b^{4}\right ) \left (2 c d x +b d \right )^{\frac {7}{2}}+16 \left (-\frac {23}{8} a^{3} c^{3} d^{2}+\frac {69}{32} a^{2} b^{2} c^{2} d^{2}-\frac {69}{128} a \,b^{4} c \,d^{2}+\frac {23}{512} d^{2} b^{6}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {\left (\frac {165}{2} a^{2} c^{2}-\frac {165}{4} c a \,b^{2}+\frac {165}{32} b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\right )\) | \(432\) |
| pseudoelliptic | \(\frac {660 d^{5} \left (-\frac {2 \left (a c -\frac {b^{2}}{4}\right ) d^{2} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (\frac {32 c^{4} x^{4}}{55}+\frac {64 x^{2} \left (b x +a \right ) c^{3}}{55}+\left (\frac {32}{55} b^{2} x^{2}+\frac {64}{55} a b x +a^{2}\right ) c^{2}-\frac {23 c a \,b^{2}}{110}+\frac {23 b^{4}}{880}\right ) \left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}{3}-\frac {157 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (-\frac {32 c^{4} x^{4}}{157}-\frac {64 x^{2} \left (b x +a \right ) c^{3}}{157}+\left (-\frac {32}{157} b^{2} x^{2}-\frac {64}{157} a b x +a^{2}\right ) c^{2}-\frac {189 c a \,b^{2}}{314}+\frac {189 b^{4}}{2512}\right ) \left (d \left (2 c x +b \right )\right )^{\frac {7}{2}}}{2310}+\sqrt {2}\, c^{2} d^{4} \left (a c -\frac {b^{2}}{4}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right )+\ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )\right )\right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (c \,x^{2}+b x +a \right )^{2}}\) | \(432\) |
Input:
int((2*c*d*x+b*d)^(17/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
64*c^2*d^5*(-4*a*c*d^2*(2*c*d*x+b*d)^(3/2)+b^2*d^2*(2*c*d*x+b*d)^(3/2)+1/7 *(2*c*d*x+b*d)^(7/2)+d^4*(16*((-27/32*a^2*c^2+27/64*c*a*b^2-27/512*b^4)*(2 *c*d*x+b*d)^(7/2)+(-23/8*a^3*c^3*d^2+69/32*a^2*b^2*c^2*d^2-69/128*a*b^4*c* d^2+23/512*d^2*b^6)*(2*c*d*x+b*d)^(3/2))/((2*c*d*x+b*d)^2+4*a*d^2*c-b^2*d^ 2)^2+1/8*(165/2*a^2*c^2-165/4*c*a*b^2+165/32*b^4)/(4*a*c*d^2-b^2*d^2)^(1/4 )*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 = 0.12 (sec) , antiderivative size = 1426, normalized size of antiderivative = 6.37 \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((2*c*d*x+b*d)^(17/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
1/14*(1155*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c ^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c^14 - 16384* a^7*c^15)*d^34)^(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(-4492125*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^ 4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*sqrt(2*c*d*x + b*d)*d^25 + 4492 125*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 + 8 960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c^14 - 16384*a^7*c^1 5)*d^34)^(3/4)) - 1155*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 22 40*a^3*b^8*c^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c ^14 - 16384*a^7*c^15)*d^34)^(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(-4492125*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^ 2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*sqrt(2*c* d*x + b*d)*d^25 + 4492125*I*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6* b^2*c^14 - 16384*a^7*c^15)*d^34)^(3/4)) - 1155*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^ 4*c^13 + 28672*a^6*b^2*c^14 - 16384*a^7*c^15)*d^34)^(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(-4492125*(b^10*c^ 6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*sqrt(2*c*d*x + b*d)*d^25 - 4492125*I*((b^14*c^8 - 28*a*...
Timed out. \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((2*c*d*x+b*d)**(17/2)/(c*x**2+b*x+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^(17/2)/(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 = 0.21 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.36 \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((2*c*d*x+b*d)^(17/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
64*(2*c*d*x + b*d)^(3/2)*b^2*c^2*d^7 - 256*(2*c*d*x + b*d)^(3/2)*a*c^3*d^7 + 64/7*(2*c*d*x + b*d)^(7/2)*c^2*d^5 - 165/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d ^2)^(3/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*a*c^3*d^7)* 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)) - 165/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d ^2)^(3/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*a*c^3*d^7)* 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)) + 165/4*(sqrt(2)*(-b^2*d^2 + 4*a*c* d^2)^(3/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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) + sqrt(-b^2*d^2 + 4*a*c*d^2)) - 165/4*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^ (3/4)*b^2*c^2*d^7 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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) + sqrt(-b^2*d^2 + 4*a*c*d^2)) + 2*(23*(2*c*d*x + b*d)^(3/2)*b^6*c^2*d^11 - 276*(2*c*d*x + b*d)^(3/2)*a*b^4*c^3*d^11 + 1104*(2*c*d*x + b*d)^(3/2)*a^2* b^2*c^4*d^11 - 1472*(2*c*d*x + b*d)^(3/2)*a^3*c^5*d^11 - 27*(2*c*d*x + b*d )^(7/2)*b^4*c^2*d^9 + 216*(2*c*d*x + b*d)^(7/2)*a*b^2*c^3*d^9 - 432*(2*c*d *x + b*d)^(7/2)*a^2*c^4*d^9)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2
Time = 0.11 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.62 \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {64\,c^2\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{7}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,\left (864\,a^2\,c^4\,d^9-432\,a\,b^2\,c^3\,d^9+54\,b^4\,c^2\,d^9\right )+{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (2944\,a^3\,c^5\,d^{11}-2208\,a^2\,b^2\,c^4\,d^{11}+552\,a\,b^4\,c^3\,d^{11}-46\,b^6\,c^2\,d^{11}\right )}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}-64\,c^2\,d^7\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (4\,a\,c-b^2\right )+165\,c^2\,d^{17/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}+c^2\,d^{17/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}\,1{}\mathrm {i}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}\,165{}\mathrm {i} \] Input:
int((b*d + 2*c*d*x)^(17/2)/(a + b*x + c*x^2)^3,x)
Output:
(64*c^2*d^5*(b*d + 2*c*d*x)^(7/2))/7 - ((b*d + 2*c*d*x)^(7/2)*(864*a^2*c^4 *d^9 + 54*b^4*c^2*d^9 - 432*a*b^2*c^3*d^9) + (b*d + 2*c*d*x)^(3/2)*(2944*a ^3*c^5*d^11 - 46*b^6*c^2*d^11 + 552*a*b^4*c^3*d^11 - 2208*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) - 64*c^2*d^7*(b*d + 2*c*d*x)^(3/2)*(4* a*c - b^2) + 165*c^2*d^(17/2)*atan(((b*d + 2*c*d*x)^(1/2)*(b^2 - 4*a*c)^(7 /4))/(d^(1/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(b^2 - 4*a*c)^(7/4) + c^2*d ^(17/2)*atan(((b*d + 2*c*d*x)^(1/2)*(b^2 - 4*a*c)^(7/4)*1i)/(d^(1/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(b^2 - 4*a*c)^(7/4)*165i
Time = 0.24 (sec) , antiderivative size = 3296, normalized size of antiderivative = 14.71 \[ \int \frac {(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((2*c*d*x+b*d)^(17/2)/(c*x^2+b*x+a)^3,x)
Output:
(sqrt(d)*d**8*( - 9240*(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**3* c**3 + 2310*(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*b**2*c**2 - 18480*(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*b*c**3*x - 18480 *(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqr t(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a**2*c**4*x**2 + 4620*(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**3*c**2*x - 4620*(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**2*c**3*x**2 - 18480*(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**4*x**3 - 9240*(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**5*x**4 + 2310*(4*a*c - b**2)**(3/4)*sqrt(2)*a tan(((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 + 4620*(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)*s qrt(2)))*b**3*c**3*x**3 + 2310*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a...