Integrand size = 19, antiderivative size = 326 \[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\frac {(c+d x) \sqrt {a+b (c+d x)^4}}{\sqrt {b} d^3 \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )}-\frac {c \text {arctanh}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a+b (c+d x)^4}}\right )}{\sqrt {b} d^3}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{b^{3/4} d^3 \sqrt {a+b (c+d x)^4}}+\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} b^{3/4} d^3 \sqrt {a+b (c+d x)^4}} \] Output:
(d*x+c)*(a+b*(d*x+c)^4)^(1/2)/b^(1/2)/d^3/(a^(1/2)+b^(1/2)*(d*x+c)^2)-c*ar ctanh(b^(1/2)*(d*x+c)^2/(a+b*(d*x+c)^4)^(1/2))/b^(1/2)/d^3-a^(1/4)*(a^(1/2 )+b^(1/2)*(d*x+c)^2)*((a+b*(d*x+c)^4)/(a^(1/2)+b^(1/2)*(d*x+c)^2)^2)^(1/2) *EllipticE(sin(2*arctan(b^(1/4)*(d*x+c)/a^(1/4))),1/2*2^(1/2))/b^(3/4)/d^3 /(a+b*(d*x+c)^4)^(1/2)+1/2*(b^(1/2)*c^2+a^(1/2))*(a^(1/2)+b^(1/2)*(d*x+c)^ 2)*((a+b*(d*x+c)^4)/(a^(1/2)+b^(1/2)*(d*x+c)^2)^2)^(1/2)*InverseJacobiAM(2 *arctan(b^(1/4)*(d*x+c)/a^(1/4)),1/2*2^(1/2))/a^(1/4)/b^(3/4)/d^3/(a+b*(d* x+c)^4)^(1/2)
Result contains complex when optimal does not.
Time = 11.67 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.70 \[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\frac {2 \left ((-1)^{3/4} a^{3/4}+i \sqrt {a} \sqrt [4]{b} (c+d x)+\sqrt [4]{-1} \sqrt [4]{a} \sqrt {b} (c+d x)^2+b^{3/4} (c+d x)^3\right )-\frac {2 i \left (\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)\right )^2 \sqrt {\frac {(1-i) \left ((-1)^{3/4} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}} \sqrt {-\frac {i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}} \sqrt {-\frac {(1+i) \left ((-1)^{3/4} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}} \left ((-1)^{3/4} \sqrt {a} E\left (\left .\arcsin \left (\sqrt {-\frac {i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}}\right )\right |-1\right )+\left (-(-1)^{3/4} \sqrt {a}+2 \sqrt [4]{a} \sqrt [4]{b} c+(-1)^{3/4} \sqrt {b} c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}}\right ),-1\right )-4 \sqrt [4]{a} \sqrt [4]{b} c \operatorname {EllipticPi}\left (-i,\arcsin \left (\sqrt {-\frac {i \left (\sqrt [4]{-1} \sqrt [4]{a}+\sqrt [4]{b} (c+d x)\right )}{\sqrt [4]{-1} \sqrt [4]{a}-\sqrt [4]{b} (c+d x)}}\right ),-1\right )\right )}{\sqrt [4]{a}}}{2 b^{3/4} d^3 \sqrt {a+b (c+d x)^4}} \] Input:
Integrate[x^2/Sqrt[a + b*(c + d*x)^4],x]
Output:
(2*((-1)^(3/4)*a^(3/4) + I*Sqrt[a]*b^(1/4)*(c + d*x) + (-1)^(1/4)*a^(1/4)* Sqrt[b]*(c + d*x)^2 + b^(3/4)*(c + d*x)^3) - ((2*I)*((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))^2*Sqrt[((1 - I)*((-1)^(3/4)*a^(1/4) - b^(1/4)*(c + d*x) ))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]*Sqrt[((-I)*((-1)^(1/4)*a^(1/4 ) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]*Sqrt[((- 1 - I)*((-1)^(3/4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^( 1/4)*(c + d*x))]*((-1)^(3/4)*Sqrt[a]*EllipticE[ArcSin[Sqrt[((-I)*((-1)^(1/ 4)*a^(1/4) + b^(1/4)*(c + d*x)))/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))] ], -1] + (-((-1)^(3/4)*Sqrt[a]) + 2*a^(1/4)*b^(1/4)*c + (-1)^(3/4)*Sqrt[b] *c^2)*EllipticF[ArcSin[Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)) )/((-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]], -1] - 4*a^(1/4)*b^(1/4)*c*El lipticPi[-I, ArcSin[Sqrt[((-I)*((-1)^(1/4)*a^(1/4) + b^(1/4)*(c + d*x)))/( (-1)^(1/4)*a^(1/4) - b^(1/4)*(c + d*x))]], -1]))/a^(1/4))/(2*b^(3/4)*d^3*S qrt[a + b*(c + d*x)^4])
Time = 0.68 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {896, 2424, 2009}
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.
| \(\displaystyle \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int \frac {d^2 x^2}{\sqrt {b (c+d x)^4+a}}d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2424 |
| \(\displaystyle \frac {\int \left (\frac {c^2+(c+d x)^2}{\sqrt {b (c+d x)^4+a}}-\frac {2 c (c+d x)}{\sqrt {b (c+d x)^4+a}}\right )d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} b^{3/4} \sqrt {a+b (c+d x)^4}}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right ) \sqrt {\frac {a+b (c+d x)^4}{\left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{b^{3/4} \sqrt {a+b (c+d x)^4}}-\frac {c \text {arctanh}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a+b (c+d x)^4}}\right )}{\sqrt {b}}+\frac {(c+d x) \sqrt {a+b (c+d x)^4}}{\sqrt {b} \left (\sqrt {a}+\sqrt {b} (c+d x)^2\right )}}{d^3}\) |
Input:
Int[x^2/Sqrt[a + b*(c + d*x)^4],x]
Output:
(((c + d*x)*Sqrt[a + b*(c + d*x)^4])/(Sqrt[b]*(Sqrt[a] + Sqrt[b]*(c + d*x) ^2)) - (c*ArcTanh[(Sqrt[b]*(c + d*x)^2)/Sqrt[a + b*(c + d*x)^4]])/Sqrt[b] - (a^(1/4)*(Sqrt[a] + Sqrt[b]*(c + d*x)^2)*Sqrt[(a + b*(c + d*x)^4)/(Sqrt[ a] + Sqrt[b]*(c + d*x)^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*(c + d*x))/a^(1/4 )], 1/2])/(b^(3/4)*Sqrt[a + b*(c + d*x)^4]) + ((Sqrt[a] + Sqrt[b]*c^2)*(Sq rt[a] + Sqrt[b]*(c + d*x)^2)*Sqrt[(a + b*(c + d*x)^4)/(Sqrt[a] + Sqrt[b]*( c + d*x)^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*(c + d*x))/a^(1/4)], 1/2])/(2*a ^(1/4)*b^(3/4)*Sqrt[a + b*(c + d*x)^4]))/d^3
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2 *((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 5.14 (sec) , antiderivative size = 2153, normalized size of antiderivative = 6.60
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2153\) |
| elliptic | \(\text {Expression too large to display}\) | \(2153\) |
Input:
int(x^2/(a+b*(d*x+c)^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
((x-(1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-I/b*(-a*b ^3)^(1/4)-c)/d)+(-(-I/b*(-a*b^3)^(1/4)-c)/d+(1/b*(-a*b^3)^(1/4)-c)/d)*(((- I/b*(-a*b^3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4)-c)/d)*(x-(1/b*(-a*b^3)^(1/4)-c )/d)/((-I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3) ^(1/4)-c)/d))^(1/2)*(x-(I/b*(-a*b^3)^(1/4)-c)/d)^2*(((I/b*(-a*b^3)^(1/4)-c )/d-(1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-1/b*(-a*b^3)^(1/4)-c)/d)/((-1/b*(-a*b^3 )^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c)/d))^(1/2) *(((I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)*(x-(-I/b*(-a*b^3)^(1 /4)-c)/d)/((-I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b^3)^(1/4)-c)/d)/(x-(I/b*(-a *b^3)^(1/4)-c)/d))^(1/2)*(((I/b*(-a*b^3)^(1/4)-c)/d^2*(1/b*(-a*b^3)^(1/4)- c)-(-I/b*(-a*b^3)^(1/4)-c)/d^2*(1/b*(-a*b^3)^(1/4)-c)+(-I/b*(-a*b^3)^(1/4) -c)/d^2*(I/b*(-a*b^3)^(1/4)-c)+(I/b*(-a*b^3)^(1/4)-c)^2/d^2)/((-I/b*(-a*b^ 3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4)-c)/d)/((I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a *b^3)^(1/4)-c)/d)*EllipticF((((-I/b*(-a*b^3)^(1/4)-c)/d-(I/b*(-a*b^3)^(1/4 )-c)/d)*(x-(1/b*(-a*b^3)^(1/4)-c)/d)/((-I/b*(-a*b^3)^(1/4)-c)/d-(1/b*(-a*b ^3)^(1/4)-c)/d)/(x-(I/b*(-a*b^3)^(1/4)-c)/d))^(1/2),(((I/b*(-a*b^3)^(1/4)- c)/d-(-1/b*(-a*b^3)^(1/4)-c)/d)*(-(-I/b*(-a*b^3)^(1/4)-c)/d+(1/b*(-a*b^3)^ (1/4)-c)/d)/((1/b*(-a*b^3)^(1/4)-c)/d-(-1/b*(-a*b^3)^(1/4)-c)/d)/((I/b*(-a *b^3)^(1/4)-c)/d-(-I/b*(-a*b^3)^(1/4)-c)/d))^(1/2))+((1/b*(-a*b^3)^(1/4)-c )/d-(-1/b*(-a*b^3)^(1/4)-c)/d)*EllipticE((((-I/b*(-a*b^3)^(1/4)-c)/d-(I...
\[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (d x + c\right )}^{4} b + a}} \,d x } \] Input:
integrate(x^2/(a+b*(d*x+c)^4)^(1/2),x, algorithm="fricas")
Output:
integral(x^2/sqrt(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d* x + b*c^4 + a), x)
\[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\int \frac {x^{2}}{\sqrt {a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4}}}\, dx \] Input:
integrate(x**2/(a+b*(d*x+c)**4)**(1/2),x)
Output:
Integral(x**2/sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c* d**3*x**3 + b*d**4*x**4), x)
\[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (d x + c\right )}^{4} b + a}} \,d x } \] Input:
integrate(x^2/(a+b*(d*x+c)^4)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2/sqrt((d*x + c)^4*b + a), x)
\[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (d x + c\right )}^{4} b + a}} \,d x } \] Input:
integrate(x^2/(a+b*(d*x+c)^4)^(1/2),x, algorithm="giac")
Output:
integrate(x^2/sqrt((d*x + c)^4*b + a), x)
Timed out. \[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\int \frac {x^2}{\sqrt {a+b\,{\left (c+d\,x\right )}^4}} \,d x \] Input:
int(x^2/(a + b*(c + d*x)^4)^(1/2),x)
Output:
int(x^2/(a + b*(c + d*x)^4)^(1/2), x)
\[ \int \frac {x^2}{\sqrt {a+b (c+d x)^4}} \, dx=\frac {\sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}-b \,c^{2}-2 b c d x -b \,d^{2} x^{2}\right ) c +2 \sqrt {b}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}-b \,c^{2}-2 b c d x -b \,d^{2} x^{2}\right ) c +2 \left (\int \frac {\sqrt {b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}}{b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}d x \right ) b \,c^{2} d +\left (\int \frac {\sqrt {b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}\, x^{2}}{b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}d x \right ) b \,d^{3}+2 \left (\int \frac {\sqrt {b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}\, x}{b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a}d x \right ) b c \,d^{2}}{b \,d^{3}} \] Input:
int(x^2/(a+b*(d*x+c)^4)^(1/2),x)
Output:
(sqrt(b)*log( - sqrt(b)*sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x** 2 + 4*b*c*d**3*x**3 + b*d**4*x**4) - b*c**2 - 2*b*c*d*x - b*d**2*x**2)*c + 2*sqrt(b)*log(sqrt(b)*sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d**3*x**3 + b*d**4*x**4) - b*c**2 - 2*b*c*d*x - b*d**2*x**2)*c + 2*int(sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d**3*x** 3 + b*d**4*x**4)/(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d **3*x**3 + b*d**4*x**4),x)*b*c**2*d + int((sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d**3*x**3 + b*d**4*x**4)*x**2)/(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d**3*x**3 + b*d**4*x**4),x)*b*d* *3 + 2*int((sqrt(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d* *3*x**3 + b*d**4*x**4)*x)/(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d**3*x**3 + b*d**4*x**4),x)*b*c*d**2)/(b*d**3)