Integrand size = 26, antiderivative size = 258 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {234 c^2}{5 \left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2}}+\frac {234 c^2}{\left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x}}-\frac {1}{2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^2}+\frac {13 c}{2 \left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )}+\frac {117 c^2 \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}}-\frac {117 c^2 \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{17/4} d^{7/2}} \] Output:
234/5*c^2/(-4*a*c+b^2)^3/d/(2*c*d*x+b*d)^(5/2)+234*c^2/(-4*a*c+b^2)^4/d^3/ (2*c*d*x+b*d)^(1/2)-1/2/(-4*a*c+b^2)/d/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^2 +13/2*c/(-4*a*c+b^2)^2/d/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)+117*c^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)^(17/4)/d^(7/2 )-117*c^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)^(17/4)/d^(7/2)
Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {\left (\frac {1}{10}+\frac {i}{10}\right ) c^2 \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) (b+2 c x) \left (-32 b^6+384 a b^4 c-1536 a^2 b^2 c^2+2048 a^3 c^3-416 b^4 (b+2 c x)^2+3328 a b^2 c (b+2 c x)^2-6656 a^2 c^2 (b+2 c x)^2+1053 b^2 (b+2 c x)^4-4212 a c (b+2 c x)^4-585 (b+2 c x)^6\right )}{c^2 \left (b^2-4 a c\right )^4 (a+x (b+c x))^2}-\frac {585 (b+2 c x)^{7/2} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}+\frac {585 (b+2 c x)^{7/2} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}-\frac {585 (b+2 c x)^{7/2} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{17/4}}\right )}{(d (b+2 c x))^{7/2}} \] Input:
Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]
Output:
((1/10 + I/10)*c^2*(((-1/8 + I/8)*(b + 2*c*x)*(-32*b^6 + 384*a*b^4*c - 153 6*a^2*b^2*c^2 + 2048*a^3*c^3 - 416*b^4*(b + 2*c*x)^2 + 3328*a*b^2*c*(b + 2 *c*x)^2 - 6656*a^2*c^2*(b + 2*c*x)^2 + 1053*b^2*(b + 2*c*x)^4 - 4212*a*c*( b + 2*c*x)^4 - 585*(b + 2*c*x)^6))/(c^2*(b^2 - 4*a*c)^4*(a + x*(b + c*x))^ 2) - (585*(b + 2*c*x)^(7/2)*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4* a*c)^(1/4)])/(b^2 - 4*a*c)^(17/4) + (585*(b + 2*c*x)^(7/2)*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(17/4) - (585*(b + 2*c*x)^(7/2)*ArcTanh[((1 + I)*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(Sqr t[b^2 - 4*a*c] + I*(b + 2*c*x))])/(b^2 - 4*a*c)^(17/4)))/(d*(b + 2*c*x))^( 7/2)
Time = 0.53 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1111, 1111, 1117, 1117, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^3 (b d+2 c d x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle -\frac {13 c \int \frac {1}{(b d+2 c x d)^{7/2} \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \int \frac {1}{(b d+2 c x d)^{7/2} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\int \frac {1}{(b d+2 c x d)^{3/2} \left (c x^2+b x+a\right )}dx}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {\int \frac {\sqrt {b d+2 c x d}}{c x^2+b x+a}dx}{d^2 \left (b^2-4 a c\right )}+\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 1118 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {\int \frac {4 c d^2 \sqrt {b d+2 c x d}}{\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2}d(b d+2 c x d)}{2 c d^3 \left (b^2-4 a c\right )}+\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {2 \int -\frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)}{d \left (b^2-4 a c\right )}+\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {2 \int \frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)}{d \left (b^2-4 a c\right )}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {4 \int \frac {b d+2 c x d}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}}{d \left (b^2-4 a c\right )}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {4 \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {1}{2} \int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}\right )}{d \left (b^2-4 a c\right )}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {4 \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {13 c \left (-\frac {9 c \left (\frac {\frac {4}{d \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {4 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )}}{d^2 \left (b^2-4 a c\right )}+\frac {4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}\right )}{b^2-4 a c}-\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}}\) |
Input:
Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]
Output:
-1/2*1/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^2) - (13*c *(-(1/((b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2))) - (9*c*(4 /(5*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)) + (4/((b^2 - 4*a*c)*d*Sqrt[b*d + 2*c*d*x]) - (4*(-1/2*ArcTan[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqr t[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])))/((b^2 - 4*a*c)*d ))/((b^2 - 4*a*c)*d^2)))/(b^2 - 4*a*c)))/(2*(b^2 - 4*a*c))
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_S ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* (b^2 - 4*a*c))), x] - Simp[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e , m}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] && !G tQ[m, 1] && RationalQ[m] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) 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] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
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]
Time = 1.54 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(64 d^{5} c^{2} \left (\frac {\frac {\frac {21 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (\frac {25}{128} a \,d^{2} c -\frac {25}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{8} \left (4 a c -b^{2}\right )^{4}}-\frac {1}{5 d^{6} \left (4 a c -b^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {3}{d^{8} \left (4 a c -b^{2}\right )^{4} \sqrt {2 c d x +b d}}\right )\) | \(382\) |
| default | \(64 d^{5} c^{2} \left (\frac {\frac {\frac {21 \left (2 c d x +b d \right )^{\frac {7}{2}}}{32}+16 \left (\frac {25}{128} a \,d^{2} c -\frac {25}{512} b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}}{d^{8} \left (4 a c -b^{2}\right )^{4}}-\frac {1}{5 d^{6} \left (4 a c -b^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {3}{d^{8} \left (4 a c -b^{2}\right )^{4} \sqrt {2 c d x +b d}}\right )\) | \(382\) |
| pseudoelliptic | \(-\frac {-\frac {585 c^{2} \sqrt {2}\, \left (2 c x +b \right )^{2} \left (c \,x^{2}+b x +a \right )^{2} \left (\ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right ) \sqrt {d \left (2 c x +b \right )}}{1024}+\left (-\frac {585 x^{6} c^{6}}{32}-\frac {1053 x^{4} \left (\frac {5 b x}{3}+a \right ) c^{5}}{32}-13 \left (\frac {297}{64} b^{2} x^{2}+\frac {81}{16} a b x +a^{2}\right ) x^{2} c^{4}+\left (a^{3}-\frac {117}{4} b^{3} x^{3}-\frac {2743}{64} a \,b^{2} x^{2}-13 a^{2} b x \right ) c^{3}-4 b^{2} \left (\frac {13 b x}{8}+a \right ) \left (\frac {221 b x}{256}+a \right ) c^{2}-\frac {125 \left (\frac {13 b x}{25}+a \right ) b^{4} c}{512}+\frac {5 b^{6}}{512}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{5 \sqrt {d \left (2 c x +b \right )}\, \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \left (2 c x +b \right )^{2} \left (a c -\frac {b^{2}}{4}\right )^{4} d^{3} \left (c \,x^{2}+b x +a \right )^{2}}\) | \(456\) |
Input:
int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
64*d^5*c^2*(1/d^8/(4*a*c-b^2)^4*(16*(21/512*(2*c*d*x+b*d)^(7/2)+(25/128*a* d^2*c-25/512*b^2*d^2)*(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+117/256/(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)))-1/5/d^6/(4*a*c-b^2)^3/(2*c*d*x+b*d)^(5/2)+3/d^8/(4*a*c-b^2)^4/(2*c* d*x+b*d)^(1/2))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 4956, normalized size of antiderivative = 19.21 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(2*c*d*x+b*d)^(7/2)/(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. 958 vs. \(2 (224) = 448\).
Time = 0.19 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.71 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
-117*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*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^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt( 2)*a^5*c^5*d^5) - 117*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*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^10*d^5 - 20*sqrt(2)*a*b^8*c*d^5 + 160*sqrt( 2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1280*sqrt(2)*a^4*b^2*c^ 4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) + 117/2*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2 *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^10*d^5 - 20*sqrt(2)*a*b^8*c*d ^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d^5 + 1280*sqrt (2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 117/2*(-b^2*d^2 + 4*a*c* d^2)^(3/4)*c^2*log(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sq rt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^10*d^5 - 20*sqr t(2)*a*b^8*c*d^5 + 160*sqrt(2)*a^2*b^6*c^2*d^5 - 640*sqrt(2)*a^3*b^4*c^3*d ^5 + 1280*sqrt(2)*a^4*b^2*c^4*d^5 - 1024*sqrt(2)*a^5*c^5*d^5) - 2*(25*(2*c *d*x + b*d)^(3/2)*b^2*c^2*d^2 - 100*(2*c*d*x + b*d)^(3/2)*a*c^3*d^2 - 21*( 2*c*d*x + b*d)^(7/2)*c^2)/((b^8*d^3 - 16*a*b^6*c*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 + 256*a^4*c^4*d^3)*(b^2*d^2 - 4*a*c*d^2 - (2*c*d*...
Time = 5.65 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.10 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {234\,c^2\,{\left (b\,d+2\,c\,d\,x\right )}^6}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3}-\frac {2106\,c^2\,{\left (b\,d+2\,c\,d\,x\right )}^4}{5\,\left (-64\,d\,a^3\,c^3+48\,d\,a^2\,b^2\,c^2-12\,d\,a\,b^4\,c+d\,b^6\right )}-\frac {64\,c^2\,d^3}{5\,\left (4\,a\,c-b^2\right )}+\frac {832\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^2}{5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4+b^4\,d^4\right )-{\left (b\,d+2\,c\,d\,x\right )}^{9/2}\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+{\left (b\,d+2\,c\,d\,x\right )}^{13/2}}+\frac {117\,c^2\,\mathrm {atan}\left (\frac {b^8\,\sqrt {b\,d+2\,c\,d\,x}+256\,a^4\,c^4\,\sqrt {b\,d+2\,c\,d\,x}+96\,a^2\,b^4\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-256\,a^3\,b^2\,c^3\,\sqrt {b\,d+2\,c\,d\,x}-16\,a\,b^6\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{17/4}}\right )}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{17/4}}+\frac {c^2\,\mathrm {atan}\left (\frac {b^8\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^4\,c^4\,\sqrt {b\,d+2\,c\,d\,x}\,256{}\mathrm {i}+a^2\,b^4\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,96{}\mathrm {i}-a^3\,b^2\,c^3\,\sqrt {b\,d+2\,c\,d\,x}\,256{}\mathrm {i}-a\,b^6\,c\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{17/4}}\right )\,117{}\mathrm {i}}{d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{17/4}} \] Input:
int(1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x)
Output:
((234*c^2*(b*d + 2*c*d*x)^6)/(b^8*d^3 + 256*a^4*c^4*d^3 + 96*a^2*b^4*c^2*d ^3 - 256*a^3*b^2*c^3*d^3 - 16*a*b^6*c*d^3) - (2106*c^2*(b*d + 2*c*d*x)^4)/ (5*(b^6*d - 64*a^3*c^3*d + 48*a^2*b^2*c^2*d - 12*a*b^4*c*d)) - (64*c^2*d^3 )/(5*(4*a*c - b^2)) + (832*c^2*d*(b*d + 2*c*d*x)^2)/(5*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/((b*d + 2*c*d*x)^(5/2)*(b^4*d^4 + 16*a^2*c^2*d^4 - 8*a*b^2*c *d^4) - (b*d + 2*c*d*x)^(9/2)*(2*b^2*d^2 - 8*a*c*d^2) + (b*d + 2*c*d*x)^(1 3/2)) + (117*c^2*atan((b^8*(b*d + 2*c*d*x)^(1/2) + 256*a^4*c^4*(b*d + 2*c* d*x)^(1/2) + 96*a^2*b^4*c^2*(b*d + 2*c*d*x)^(1/2) - 256*a^3*b^2*c^3*(b*d + 2*c*d*x)^(1/2) - 16*a*b^6*c*(b*d + 2*c*d*x)^(1/2))/(d^(1/2)*(b^2 - 4*a*c) ^(17/4))))/(d^(7/2)*(b^2 - 4*a*c)^(17/4)) + (c^2*atan((b^8*(b*d + 2*c*d*x) ^(1/2)*1i + a^4*c^4*(b*d + 2*c*d*x)^(1/2)*256i + a^2*b^4*c^2*(b*d + 2*c*d* x)^(1/2)*96i - a^3*b^2*c^3*(b*d + 2*c*d*x)^(1/2)*256i - a*b^6*c*(b*d + 2*c *d*x)^(1/2)*16i)/(d^(1/2)*(b^2 - 4*a*c)^(17/4)))*117i)/(d^(7/2)*(b^2 - 4*a *c)^(17/4))
Time = 0.36 (sec) , antiderivative size = 4318, normalized size of antiderivative = 16.74 \[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x)
Output:
(sqrt(d)*( - 1170*sqrt(b + 2*c*x)*(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 - 4680*sqrt(b + 2*c*x)*(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 - 4680*sqrt(b + 2*c*x)*(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**4*x**2 - 2340*sqrt(b + 2*c*x)*(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 - 11700*sqrt(b + 2*c*x)*( 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 - 18720*sqrt (b + 2*c*x)*(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 - 9360*sqrt(b + 2*c*x)*(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 - 1170*sqrt(b + 2*c*x)*(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**4*c**2*x**2 - 7020*sqrt(b + 2*c*x)*(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**3*c**3*x**3 - 15210*sqrt(b + 2*c*x)*(4*a*c - b**2)**(3/4)*...