Integrand size = 24, antiderivative size = 613 \[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {9 c d^2 \sqrt [3]{e x}}{b e}+\frac {3 d^3 (e x)^{4/3}}{4 b e^2}+\frac {c \left (b c^2-3 a d^2\right ) \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{5/6} b^{7/6} e^{2/3}}-\frac {\left (b^{3/2} c^3+3 \sqrt {3} \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2-\sqrt {3} a^{3/2} d^3\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{5/6} b^{5/3} e^{2/3}}+\frac {\left (b^{3/2} c^3-3 \sqrt {3} \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2+\sqrt {3} a^{3/2} d^3\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{5/6} b^{5/3} e^{2/3}}-\frac {d \left (3 b c^2-a d^2\right ) \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [3]{a} b^{5/3} e^{2/3}}-\frac {\left (\sqrt {3} b^{3/2} c^3-3 \sqrt {a} b c^2 d-3 \sqrt {3} a \sqrt {b} c d^2+a^{3/2} d^3\right ) \log \left (\sqrt [3]{a} e^{2/3}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{5/6} b^{5/3} e^{2/3}}+\frac {\left (\sqrt {3} b^{3/2} c^3+3 \sqrt {a} b c^2 d-3 \sqrt {3} a \sqrt {b} c d^2-a^{3/2} d^3\right ) \log \left (\sqrt [3]{a} e^{2/3}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{5/6} b^{5/3} e^{2/3}} \] Output:
9*c*d^2*(e*x)^(1/3)/b/e+3/4*d^3*(e*x)^(4/3)/b/e^2+c*(-3*a*d^2+b*c^2)*arcta n(b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/b^(7/6)/e^(2/3)+1/2*(b^(3/2 )*c^3+3*3^(1/2)*a^(1/2)*b*c^2*d-3*a*b^(1/2)*c*d^2-3^(1/2)*a^(3/2)*d^3)*arc tan(-3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/b^(5/3)/e^(2/3 )+1/2*(b^(3/2)*c^3-3*3^(1/2)*a^(1/2)*b*c^2*d-3*a*b^(1/2)*c*d^2+3^(1/2)*a^( 3/2)*d^3)*arctan(3^(1/2)+2*b^(1/6)*(e*x)^(1/3)/a^(1/6)/e^(1/3))/a^(5/6)/b^ (5/3)/e^(2/3)-1/2*d*(-a*d^2+3*b*c^2)*ln(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3 ))/a^(1/3)/b^(5/3)/e^(2/3)-1/4*(3^(1/2)*b^(3/2)*c^3-3*a^(1/2)*b*c^2*d-3*3^ (1/2)*a*b^(1/2)*c*d^2+a^(3/2)*d^3)*ln(a^(1/3)*e^(2/3)-3^(1/2)*a^(1/6)*b^(1 /6)*e^(1/3)*(e*x)^(1/3)+b^(1/3)*(e*x)^(2/3))/a^(5/6)/b^(5/3)/e^(2/3)+1/4*( 3^(1/2)*b^(3/2)*c^3+3*a^(1/2)*b*c^2*d-3*3^(1/2)*a*b^(1/2)*c*d^2-a^(3/2)*d^ 3)*ln(a^(1/3)*e^(2/3)+3^(1/2)*a^(1/6)*b^(1/6)*e^(1/3)*(e*x)^(1/3)+b^(1/3)* (e*x)^(2/3))/a^(5/6)/b^(5/3)/e^(2/3)
Time = 0.80 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.78 \[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\frac {x^{2/3} \left (3 a^{5/6} b^{2/3} d^2 \sqrt [3]{x} (12 c+d x)-2 \left (b^{3/2} c^3+3 \sqrt {3} \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2-\sqrt {3} a^{3/2} d^3\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 \left (b^{3/2} c^3-3 \sqrt {3} \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2+\sqrt {3} a^{3/2} d^3\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+4 \sqrt {b} c \left (b c^2-3 a d^2\right ) \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{x}}{\sqrt [6]{a}}\right )+2 \sqrt {a} \left (-3 b c^2 d+a d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{2/3}\right )-\left (\sqrt {3} b^{3/2} c^3-3 \sqrt {a} b c^2 d-3 \sqrt {3} a \sqrt {b} c d^2+a^{3/2} d^3\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )+\left (\sqrt {3} b^{3/2} c^3+3 \sqrt {a} b c^2 d-3 \sqrt {3} a \sqrt {b} c d^2-a^{3/2} d^3\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{x}+\sqrt [3]{b} x^{2/3}\right )\right )}{4 a^{5/6} b^{5/3} (e x)^{2/3}} \] Input:
Integrate[(c + d*x)^3/((e*x)^(2/3)*(a + b*x^2)),x]
Output:
(x^(2/3)*(3*a^(5/6)*b^(2/3)*d^2*x^(1/3)*(12*c + d*x) - 2*(b^(3/2)*c^3 + 3* Sqrt[3]*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 - Sqrt[3]*a^(3/2)*d^3)*ArcTan[ Sqrt[3] - (2*b^(1/6)*x^(1/3))/a^(1/6)] + 2*(b^(3/2)*c^3 - 3*Sqrt[3]*Sqrt[a ]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 + Sqrt[3]*a^(3/2)*d^3)*ArcTan[Sqrt[3] + (2*b ^(1/6)*x^(1/3))/a^(1/6)] + 4*Sqrt[b]*c*(b*c^2 - 3*a*d^2)*ArcTan[(b^(1/6)*x ^(1/3))/a^(1/6)] + 2*Sqrt[a]*(-3*b*c^2*d + a*d^3)*Log[a^(1/3) + b^(1/3)*x^ (2/3)] - (Sqrt[3]*b^(3/2)*c^3 - 3*Sqrt[a]*b*c^2*d - 3*Sqrt[3]*a*Sqrt[b]*c* d^2 + a^(3/2)*d^3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1/3) *x^(2/3)] + (Sqrt[3]*b^(3/2)*c^3 + 3*Sqrt[a]*b*c^2*d - 3*Sqrt[3]*a*Sqrt[b] *c*d^2 - a^(3/2)*d^3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x^(1/3) + b^(1 /3)*x^(2/3)]))/(4*a^(5/6)*b^(5/3)*(e*x)^(2/3))
Time = 2.37 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {559, 27, 2333, 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.
| \(\displaystyle \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle \frac {3 \int \frac {4 \left (b c^3+3 b d^2 x^2 c+d \left (3 b c^2-a d^2\right ) x\right )}{3 (e x)^{2/3} \left (b x^2+a\right )}dx}{4 b}+\frac {3 d^3 (e x)^{4/3}}{4 b e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b c^3+3 b d^2 x^2 c+d \left (3 b c^2-a d^2\right ) x}{(e x)^{2/3} \left (b x^2+a\right )}dx}{b}+\frac {3 d^3 (e x)^{4/3}}{4 b e^2}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\int \left (\frac {3 c d^2}{(e x)^{2/3}}+\frac {b c^3-3 a d^2 c+d \left (3 b c^2-a d^2\right ) x}{(e x)^{2/3} \left (b x^2+a\right )}\right )dx}{b}+\frac {3 d^3 (e x)^{4/3}}{4 b e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c \left (b c^2-3 a d^2\right ) \arctan \left (\frac {\sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{a^{5/6} \sqrt [6]{b} e^{2/3}}-\frac {c \left (b c^2-3 a d^2\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}\right )}{2 a^{5/6} \sqrt [6]{b} e^{2/3}}+\frac {c \left (b c^2-3 a d^2\right ) \arctan \left (\frac {2 \sqrt [6]{b} \sqrt [3]{e x}}{\sqrt [6]{a} \sqrt [3]{e}}+\sqrt {3}\right )}{2 a^{5/6} \sqrt [6]{b} e^{2/3}}+\frac {d \left (3 b c^2-a d^2\right ) \log \left (a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}\right )}{4 \sqrt [3]{a} b^{2/3} e^{2/3}}-\frac {\sqrt {3} c \left (b c^2-3 a d^2\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{5/6} \sqrt [6]{b} e^{2/3}}+\frac {\sqrt {3} c \left (b c^2-3 a d^2\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt [3]{e} \sqrt [3]{e x}+\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{4 a^{5/6} \sqrt [6]{b} e^{2/3}}-\frac {\sqrt {3} d \left (3 b c^2-a d^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{a} b^{2/3} e^{2/3}}-\frac {d \left (3 b c^2-a d^2\right ) \log \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{2 \sqrt [3]{a} b^{2/3} e^{2/3}}+\frac {9 c d^2 \sqrt [3]{e x}}{e}}{b}+\frac {3 d^3 (e x)^{4/3}}{4 b e^2}\) |
Input:
Int[(c + d*x)^3/((e*x)^(2/3)*(a + b*x^2)),x]
Output:
(3*d^3*(e*x)^(4/3))/(4*b*e^2) + ((9*c*d^2*(e*x)^(1/3))/e + (c*(b*c^2 - 3*a *d^2)*ArcTan[(b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))])/(a^(5/6)*b^(1/6)*e^ (2/3)) - (c*(b*c^2 - 3*a*d^2)*ArcTan[Sqrt[3] - (2*b^(1/6)*(e*x)^(1/3))/(a^ (1/6)*e^(1/3))])/(2*a^(5/6)*b^(1/6)*e^(2/3)) + (c*(b*c^2 - 3*a*d^2)*ArcTan [Sqrt[3] + (2*b^(1/6)*(e*x)^(1/3))/(a^(1/6)*e^(1/3))])/(2*a^(5/6)*b^(1/6)* e^(2/3)) - (Sqrt[3]*d*(3*b*c^2 - a*d^2)*ArcTan[(1 - (2*b^(1/3)*(e*x)^(2/3) )/(a^(1/3)*e^(2/3)))/Sqrt[3]])/(2*a^(1/3)*b^(2/3)*e^(2/3)) - (d*(3*b*c^2 - a*d^2)*Log[a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)])/(2*a^(1/3)*b^(2/3)*e^( 2/3)) - (Sqrt[3]*c*(b*c^2 - 3*a*d^2)*Log[a^(1/3)*e^(2/3) - Sqrt[3]*a^(1/6) *b^(1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(4*a^(5/6)*b^(1/6)*e^ (2/3)) + (Sqrt[3]*c*(b*c^2 - 3*a*d^2)*Log[a^(1/3)*e^(2/3) + Sqrt[3]*a^(1/6 )*b^(1/6)*e^(1/3)*(e*x)^(1/3) + b^(1/3)*(e*x)^(2/3)])/(4*a^(5/6)*b^(1/6)*e ^(2/3)) + (d*(3*b*c^2 - a*d^2)*Log[a^(2/3)*e^(4/3) - a^(1/3)*b^(1/3)*e^(2/ 3)*(e*x)^(2/3) + b^(2/3)*(e*x)^(4/3)])/(4*a^(1/3)*b^(2/3)*e^(2/3)))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1)) Int[(e*x)^m*(a + b* x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 )*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && IGtQ[n, 1] && !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.86 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.69
| method | result | size |
| pseudoelliptic | \(\frac {-3 e \left (a \,d^{2}-\frac {b \,c^{2}}{3}\right ) \left (\frac {\left (-\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )+\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )\right ) \sqrt {3}}{2}+\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )+2 \arctan \left (\frac {\left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )-\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )\right ) c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+\left (18 d \left (\frac {d x}{12}+c \right ) e a \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {2}{3}} \left (a \,d^{2}-3 b \,c^{2}\right ) \left (\left (\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}+2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )+\arctan \left (\frac {\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}-2 \left (e x \right )^{\frac {1}{3}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}}}\right )\right ) \sqrt {3}+\ln \left (\left (e x \right )^{\frac {2}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}-\left (e x \right )^{\frac {2}{3}}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2}-\frac {\ln \left (\left (e x \right )^{\frac {2}{3}}+\sqrt {3}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{6}} \left (e x \right )^{\frac {1}{3}}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{3}}\right )}{2}\right )\right ) d}{2 a b \,e^{2}}\) | \(426\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1036\) |
| default | \(\text {Expression too large to display}\) | \(1036\) |
Input:
int((d*x+c)^3/(e*x)^(2/3)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2*(-3*e*(a*d^2-1/3*b*c^2)*(1/2*(-ln(3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)- (e*x)^(2/3)-(a*e^2/b)^(1/3))+ln((e*x)^(2/3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^ (1/3)+(a*e^2/b)^(1/3)))*3^(1/2)+arctan((3^(1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1 /3))/(a*e^2/b)^(1/6))+2*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))-arctan((3^(1/2 )*(a*e^2/b)^(1/6)-2*(e*x)^(1/3))/(a*e^2/b)^(1/6)))*c*(a*e^2/b)^(1/6)+(18*d *(1/12*d*x+c)*e*a*(e*x)^(1/3)+(a*e^2/b)^(2/3)*(a*d^2-3*b*c^2)*((arctan((3^ (1/2)*(a*e^2/b)^(1/6)+2*(e*x)^(1/3))/(a*e^2/b)^(1/6))+arctan((3^(1/2)*(a*e ^2/b)^(1/6)-2*(e*x)^(1/3))/(a*e^2/b)^(1/6)))*3^(1/2)+ln((e*x)^(2/3)+(a*e^2 /b)^(1/3))-1/2*ln(3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)-(e*x)^(2/3)-(a*e^2/b )^(1/3))-1/2*ln((e*x)^(2/3)+3^(1/2)*(a*e^2/b)^(1/6)*(e*x)^(1/3)+(a*e^2/b)^ (1/3))))*d)/a/b/e^2
Leaf count of result is larger than twice the leaf count of optimal. 5058 vs. \(2 (445) = 890\).
Time = 8.69 (sec) , antiderivative size = 5058, normalized size of antiderivative = 8.25 \[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3/(e*x)^(2/3)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\int \frac {\left (c + d x\right )^{3}}{\left (e x\right )^{\frac {2}{3}} \left (a + b x^{2}\right )}\, dx \] Input:
integrate((d*x+c)**3/(e*x)**(2/3)/(b*x**2+a),x)
Output:
Integral((c + d*x)**3/((e*x)**(2/3)*(a + b*x**2)), x)
Exception generated. \[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^3/(e*x)^(2/3)/(b*x^2+a),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(e>0)', see `assume?` for more de tails)Is e
Time = 0.16 (sec) , antiderivative size = 586, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^3/(e*x)^(2/3)/(b*x^2+a),x, algorithm="giac")
Output:
1/2*(b^4*c^3*e + 3*sqrt(3)*sqrt(a*b)*b^3*c^2*d*e - 3*a*b^3*c*d^2*e + sqrt( 3)*sqrt(a*b)*a*b^2*d^3*e)*arctan((sqrt(3)*(a*e^2/b)^(1/6) + 2*(e*x)^(1/3)) /(a*e^2/b)^(1/6))/(a*b^5*e^2)^(5/6) + 1/2*(b^4*c^3*e + 3*sqrt(3)*sqrt(a*b) *b^3*c^2*d*e - 3*a*b^3*c*d^2*e + sqrt(3)*sqrt(a*b)*a*b^2*d^3*e)*arctan(-(s qrt(3)*(a*e^2/b)^(1/6) - 2*(e*x)^(1/3))/(a*e^2/b)^(1/6))/(a*b^5*e^2)^(5/6) + 1/4*(sqrt(3)*b^4*c^3*e - 3*sqrt(3)*a*b^3*c*d^2*e + 3*sqrt(a*b)*b^3*c^2* d*e + sqrt(a*b)*a*b^2*d^3*e)*log(sqrt(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e* x)^(2/3) + (a*e^2/b)^(1/3))/(a*b^5*e^2)^(5/6) - 1/4*(sqrt(3)*b^4*c^3*e - 3 *sqrt(3)*a*b^3*c*d^2*e + 3*sqrt(a*b)*b^3*c^2*d*e - sqrt(a*b)*a*b^2*d^3*e)* log(-sqrt(3)*(a*e^2/b)^(1/6)*(e*x)^(1/3) + (e*x)^(2/3) + (a*e^2/b)^(1/3))/ (a*b^5*e^2)^(5/6) + ((a*b^5*e^2)^(1/6)*b*c^3 - 3*(a*b^5*e^2)^(1/6)*a*c*d^2 )*arctan((e*x)^(1/3)/(a*e^2/b)^(1/6))/(a*b^2*e) - 1/2*(3*(a*b^5*e^2)^(1/6) *b*c^2*d*abs(b)*abs(e) - (a*b^5*e^2)^(1/6)*a*d^3*abs(b)*abs(e))*log((e*x)^ (2/3) + (a*e^2/b)^(1/3))/(sqrt(a*b)*b^3*e^2) + 3/4*((e*x)^(1/3)*b^3*d^3*e^ 7*x + 12*(e*x)^(1/3)*b^3*c*d^2*e^7)/(b^4*e^8)
Time = 11.90 (sec) , antiderivative size = 3934, normalized size of antiderivative = 6.42 \[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((c + d*x)^3/((e*x)^(2/3)*(a + b*x^2)),x)
Output:
log(a^2*b^7*c^3*(e*x)^(1/3) + a*d^3*(e*x)^(1/3)*(-a^5*b^11)^(1/2) + a^3*b^ 7*e*((b^4*c^9*(-a^5*b^11)^(1/2) + a^7*b^5*d^9 + 9*a^3*b^9*c^8*d - 84*a^4*b ^8*c^6*d^3 + 126*a^5*b^7*c^4*d^5 - 36*a^6*b^6*c^2*d^7 + 9*a^4*c*d^8*(-a^5* b^11)^(1/2) - 36*a*b^3*c^7*d^2*(-a^5*b^11)^(1/2) - 84*a^3*b*c^3*d^6*(-a^5* b^11)^(1/2) + 126*a^2*b^2*c^5*d^4*(-a^5*b^11)^(1/2))/(a^5*b^10*e^2))^(1/3) - 3*a^3*b^6*c*d^2*(e*x)^(1/3) - 3*b*c^2*d*(e*x)^(1/3)*(-a^5*b^11)^(1/2))* ((b^4*c^9*(-a^5*b^11)^(1/2) + a^7*b^5*d^9 + 9*a^3*b^9*c^8*d - 84*a^4*b^8*c ^6*d^3 + 126*a^5*b^7*c^4*d^5 - 36*a^6*b^6*c^2*d^7 + 9*a^4*c*d^8*(-a^5*b^11 )^(1/2) - 36*a*b^3*c^7*d^2*(-a^5*b^11)^(1/2) - 84*a^3*b*c^3*d^6*(-a^5*b^11 )^(1/2) + 126*a^2*b^2*c^5*d^4*(-a^5*b^11)^(1/2))/(8*a^5*b^10*e^2))^(1/3) + log(a^2*b^7*c^3*(e*x)^(1/3) - a*d^3*(e*x)^(1/3)*(-a^5*b^11)^(1/2) + a^3*b ^7*e*(-(b^4*c^9*(-a^5*b^11)^(1/2) - a^7*b^5*d^9 - 9*a^3*b^9*c^8*d + 84*a^4 *b^8*c^6*d^3 - 126*a^5*b^7*c^4*d^5 + 36*a^6*b^6*c^2*d^7 + 9*a^4*c*d^8*(-a^ 5*b^11)^(1/2) - 36*a*b^3*c^7*d^2*(-a^5*b^11)^(1/2) - 84*a^3*b*c^3*d^6*(-a^ 5*b^11)^(1/2) + 126*a^2*b^2*c^5*d^4*(-a^5*b^11)^(1/2))/(a^5*b^10*e^2))^(1/ 3) - 3*a^3*b^6*c*d^2*(e*x)^(1/3) + 3*b*c^2*d*(e*x)^(1/3)*(-a^5*b^11)^(1/2) )*(-(b^4*c^9*(-a^5*b^11)^(1/2) - a^7*b^5*d^9 - 9*a^3*b^9*c^8*d + 84*a^4*b^ 8*c^6*d^3 - 126*a^5*b^7*c^4*d^5 + 36*a^6*b^6*c^2*d^7 + 9*a^4*c*d^8*(-a^5*b ^11)^(1/2) - 36*a*b^3*c^7*d^2*(-a^5*b^11)^(1/2) - 84*a^3*b*c^3*d^6*(-a^5*b ^11)^(1/2) + 126*a^2*b^2*c^5*d^4*(-a^5*b^11)^(1/2))/(8*a^5*b^10*e^2))^(...
Time = 0.23 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.13 \[ \int \frac {(c+d x)^3}{(e x)^{2/3} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^3/(e*x)^(2/3)/(b*x^2+a),x)
Output:
(6*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3))/ (b**(1/6)*a**(1/6)))*a*c*d**2 - 2*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1/6)* sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*c**3 + 2*sqrt(3)*ata n((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a **2*d**3 - 6*sqrt(3)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*x**(1/3)*b**(1/3) )/(b**(1/6)*a**(1/6)))*a*b*c**2*d - 6*sqrt(b)*sqrt(a)*atan((b**(1/6)*a**(1 /6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*c*d**2 + 2*sqrt( b)*sqrt(a)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6 )*a**(1/6)))*b*c**3 + 2*sqrt(3)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*x**(1/ 3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a**2*d**3 - 6*sqrt(3)*atan((b**(1/6)*a** (1/6)*sqrt(3) + 2*x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*b*c**2*d - 12* sqrt(b)*sqrt(a)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*a*c*d**2 + 4 *sqrt(b)*sqrt(a)*atan((x**(1/3)*b**(1/3))/(b**(1/6)*a**(1/6)))*b*c**3 + 3* sqrt(b)*sqrt(a)*sqrt(3)*log( - x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/ 3) + x**(2/3)*b**(1/3))*a*c*d**2 - sqrt(b)*sqrt(a)*sqrt(3)*log( - x**(1/3) *b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*b*c**3 - 3*sqrt (b)*sqrt(a)*sqrt(3)*log(x**(1/3)*b**(1/6)*a**(1/6)*sqrt(3) + a**(1/3) + x* *(2/3)*b**(1/3))*a*c*d**2 + sqrt(b)*sqrt(a)*sqrt(3)*log(x**(1/3)*b**(1/6)* a**(1/6)*sqrt(3) + a**(1/3) + x**(2/3)*b**(1/3))*b*c**3 + 36*x**(1/3)*b**( 2/3)*a**(1/3)*a*c*d**2 + 3*x**(1/3)*b**(2/3)*a**(1/3)*a*d**3*x + 2*log(...