Integrand size = 26, antiderivative size = 117 \[ \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx=4 d \sqrt {b d+2 c d x}-2 \sqrt [4]{b^2-4 a c} d^{3/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \sqrt [4]{b^2-4 a c} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
-2*(-4*a*c+b^2)^(1/4)*d^(3/2)*arctan((d*(2*c*x+b))^(1/2)/(-4*a*c+b^2)^(1/4 )/d^(1/2))-2*(-4*a*c+b^2)^(1/4)*d^(3/2)*arctanh((d*(2*c*x+b))^(1/2)/(-4*a* c+b^2)^(1/4)/d^(1/2))+4*d*(2*c*d*x+b*d)^(1/2)
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.64 \[ \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx=\frac {(1-i) (d (b+2 c x))^{3/2} \left ((2+2 i) \sqrt {b+2 c x}+\sqrt [4]{b^2-4 a c} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-\sqrt [4]{b^2-4 a c} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-\sqrt [4]{b^2-4 a c} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{3/2}} \]
((1 - I)*(d*(b + 2*c*x))^(3/2)*((2 + 2*I)*Sqrt[b + 2*c*x] + (b^2 - 4*a*c)^ (1/4)*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)] - (b^2 - 4 *a*c)^(1/4)*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)] - (b ^2 - 4*a*c)^(1/4)*ArcTanh[((1 + I)*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(S qrt[b^2 - 4*a*c] + I*(b + 2*c*x))]))/(b + 2*c*x)^(3/2)
Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {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.
\(\displaystyle \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \left (c x^2+b x+a\right )}dx+4 d \sqrt {b d+2 c d x}\) |
\(\Big \downarrow \) 1118 |
\(\displaystyle \frac {d \left (b^2-4 a c\right ) \int \frac {4 c d^2}{\sqrt {b d+2 c x d} \left (\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2\right )}d(b d+2 c x d)}{2 c}+4 d \sqrt {b d+2 c d x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 d^3 \left (b^2-4 a c\right ) \int -\frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)+4 d \sqrt {b d+2 c d x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 4 d \sqrt {b d+2 c d x}-2 d^3 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)\) |
\(\Big \downarrow \) 266 |
\(\displaystyle 4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \int \frac {1}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle 4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )\) |
4*d*Sqrt[b*d + 2*c*d*x] - 4*(b^2 - 4*a*c)*d^3*(ArcTan[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[S qrt[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)))
3.13.89.3.1 Defintions of rubi rules used
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_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. \(254\) vs. \(2(95)=190\).
Time = 2.65 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.18
method | result | size |
derivativedivides | \(4 d \left (\sqrt {2 c d x +b d}-\frac {d^{2} \left (4 a c -b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(255\) |
default | \(4 d \left (\sqrt {2 c d x +b d}-\frac {d^{2} \left (4 a c -b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(255\) |
pseudoelliptic | \(-\frac {2 d \left (-2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}+\frac {\sqrt {2}\, d^{2} \left (4 a c -b^{2}\right ) \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right )}{4}\right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}\) | \(286\) |
4*d*((2*c*d*x+b*d)^(1/2)-1/8*d^2*(4*a*c-b^2)/(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 = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.71 \[ \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx=4 \, \sqrt {2 \, c d x + b d} d - \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, c d x + b d} d + \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}}\right ) - i \, \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, c d x + b d} d + i \, \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}}\right ) + i \, \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, c d x + b d} d - i \, \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}}\right ) + \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}} \log \left (\sqrt {2 \, c d x + b d} d - \left ({\left (b^{2} - 4 \, a c\right )} d^{6}\right )^{\frac {1}{4}}\right ) \]
4*sqrt(2*c*d*x + b*d)*d - ((b^2 - 4*a*c)*d^6)^(1/4)*log(sqrt(2*c*d*x + b*d )*d + ((b^2 - 4*a*c)*d^6)^(1/4)) - I*((b^2 - 4*a*c)*d^6)^(1/4)*log(sqrt(2* c*d*x + b*d)*d + I*((b^2 - 4*a*c)*d^6)^(1/4)) + I*((b^2 - 4*a*c)*d^6)^(1/4 )*log(sqrt(2*c*d*x + b*d)*d - I*((b^2 - 4*a*c)*d^6)^(1/4)) + ((b^2 - 4*a*c )*d^6)^(1/4)*log(sqrt(2*c*d*x + b*d)*d - ((b^2 - 4*a*c)*d^6)^(1/4))
\[ \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx=\int \frac {\left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}}}{a + b x + c x^{2}}\, dx \]
Exception generated. \[ \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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. 354 vs. \(2 (95) = 190\).
Time = 0.29 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.03 \[ \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx=-\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} d \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {1}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} d \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + 4 \, \sqrt {2 \, c d x + b d} d \]
-sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*d*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*d^2 + 4*a*c*d^2)^(1/4)*d*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)) - 1/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*d*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)) + 1/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*d*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)) + 4*sqrt(2*c*d*x + b*d)*d
Time = 9.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83 \[ \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx=4\,d\,\sqrt {b\,d+2\,c\,d\,x}-2\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}-2\,d^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4} \]