Integrand size = 64, antiderivative size = 44 \[ \int \frac {4 a q x-3 b p x^2+a p x^4}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {q+p x^3}}{\sqrt {c} \left (b+a x^2\right )}\right )}{\sqrt {c} \sqrt {d}} \]
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 16.84 (sec) , antiderivative size = 8031, normalized size of antiderivative = 182.52 \[ \int \frac {4 a q x-3 b p x^2+a p x^4}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right )} \, dx=\text {Result too large to show} \]
Integrate[(4*a*q*x - 3*b*p*x^2 + a*p*x^4)/(Sqrt[q + p*x^3]*(b^2*c + d*q + 2*a*b*c*x^2 + d*p*x^3 + a^2*c*x^4)),x]
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.
| \(\displaystyle \int \frac {a p x^4+4 a q x-3 b p x^2}{\sqrt {p x^3+q} \left (a^2 c x^4+2 a b c x^2+b^2 c+d p x^3+d q\right )} \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int \frac {x \left (a p x^3+4 a q-3 b p x\right )}{\sqrt {p x^3+q} \left (a^2 c x^4+2 a b c x^2+b^2 c+d p x^3+d q\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {p}{a c \sqrt {p x^3+q}}-\frac {-4 a^2 c q x+5 a b c p x^2+p \left (b^2 c+d q\right )+d p^2 x^3}{a c \sqrt {p x^3+q} \left (a^2 c x^4+2 a b c x^2+b^2 c+d p x^3+d q\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d p^2 \int \frac {x^3}{\sqrt {p x^3+q} \left (a^2 c x^4+d p x^3+2 a b c x^2+b^2 c+d q\right )}dx}{a c}-\frac {p \left (b^2 c+d q\right ) \int \frac {1}{\sqrt {p x^3+q} \left (a^2 c x^4+d p x^3+2 a b c x^2+b^2 c+d q\right )}dx}{a c}-5 b p \int \frac {x^2}{\sqrt {p x^3+q} \left (a^2 c x^4+d p x^3+2 a b c x^2+b^2 c+d q\right )}dx+4 a q \int \frac {x}{\sqrt {p x^3+q} \left (a^2 c x^4+d p x^3+2 a b c x^2+b^2 c+d q\right )}dx+\frac {2 \sqrt {2+\sqrt {3}} p^{2/3} \left (\sqrt [3]{p} x+\sqrt [3]{q}\right ) \sqrt {\frac {p^{2/3} x^2-\sqrt [3]{p} \sqrt [3]{q} x+q^{2/3}}{\left (\sqrt [3]{p} x+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{p} x+\left (1-\sqrt {3}\right ) \sqrt [3]{q}}{\sqrt [3]{p} x+\left (1+\sqrt {3}\right ) \sqrt [3]{q}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} a c \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{p} x+\sqrt [3]{q}\right )}{\left (\sqrt [3]{p} x+\left (1+\sqrt {3}\right ) \sqrt [3]{q}\right )^2}} \sqrt {p x^3+q}}\) |
Int[(4*a*q*x - 3*b*p*x^2 + a*p*x^4)/(Sqrt[q + p*x^3]*(b^2*c + d*q + 2*a*b* c*x^2 + d*p*x^3 + a^2*c*x^4)),x]
3.6.72.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.74 (sec) , antiderivative size = 3170, normalized size of antiderivative = 72.05
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3170\) |
| elliptic | \(\text {Expression too large to display}\) | \(3170\) |
int((a*p*x^4-3*b*p*x^2+4*a*q*x)/(p*x^3+q)^(1/2)/(a^2*c*x^4+2*a*b*c*x^2+d*p *x^3+b^2*c+d*q),x,method=_RETURNVERBOSE)
-2/3*I/a/c*3^(1/2)*(-q*p^2)^(1/3)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/2*I*3^(1/2) /p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2)*((x-1/p*(-q*p^2)^(1/3)) /(-3/2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3)))^(1/2)*(-I*(x+1/2/ p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3)) ^(1/2)/(p*x^3+q)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/ 2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2),(I*3^(1/2)/p *(-q*p^2)^(1/3)/(-3/2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3)))^(1 /2))-I/a/c^2/p^2*2^(1/2)*sum((_alpha^3*d*p^2+5*_alpha^2*a*b*c*p-4*_alpha*a ^2*c*q+b^2*c*p+d*p*q)/_alpha/(4*_alpha^2*a^2*c+3*_alpha*d*p+4*a*b*c)/(a^6* q^4+2*a^3*b^3*p^2*q^2+b^6*p^4)*(-q*p^2)^(1/3)*(1/2*I*p*(2*x+1/p*(-I*3^(1/2 )*(-q*p^2)^(1/3)+(-q*p^2)^(1/3)))/(-q*p^2)^(1/3))^(1/2)*(p*(x-1/p*(-q*p^2) ^(1/3))/(-3*(-q*p^2)^(1/3)+I*3^(1/2)*(-q*p^2)^(1/3)))^(1/2)*(-1/2*I*p*(2*x +1/p*(I*3^(1/2)*(-q*p^2)^(1/3)+(-q*p^2)^(1/3)))/(-q*p^2)^(1/3))^(1/2)/(p*x ^3+q)^(1/2)*(-I*(-q*p^2)^(1/3)*p^5*3^(1/2)*_alpha*b^4*d-3*(-q*p^2)^(1/3)*_ alpha^3*a^4*b^2*c*p^3*q-3*(-q*p^2)^(2/3)*_alpha^2*a^4*b^2*c*p^2*q-2*(-q*p^ 2)^(1/3)*_alpha^2*a^5*b*c*p^2*q^2-3*(-q*p^2)^(1/3)*_alpha^2*a^2*b^2*d*p^4* q-4*(-q*p^2)^(1/3)*_alpha*a^3*b^3*c*p^3*q-3*(-q*p^2)^(2/3)*_alpha*a^2*b^2* d*p^3*q-4*(-q*p^2)^(2/3)*a^3*b^3*c*p^2*q-2*(-q*p^2)^(1/3)*_alpha*a^3*b*d*p ^3*q^2+I*(-q*p^2)^(1/3)*3^(1/2)*a^4*b^2*q^2*p^2*c+(-q*p^2)^(1/3)*_alpha*b^ 4*d*p^5-(-q*p^2)^(1/3)*a^4*b^2*c*p^2*q^2-2*(-q*p^2)^(2/3)*a^3*b*d*p^2*q...
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (34) = 68\).
Time = 2.29 (sec) , antiderivative size = 471, normalized size of antiderivative = 10.70 \[ \int \frac {4 a q x-3 b p x^2+a p x^4}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right )} \, dx=\left [-\frac {\sqrt {-c d} \log \left (\frac {a^{4} c^{2} x^{8} - 6 \, a^{2} c d p x^{7} - 12 \, a b c d p x^{5} + {\left (4 \, a^{3} b c^{2} + d^{2} p^{2}\right )} x^{6} + b^{4} c^{2} - 6 \, b^{2} c d q + 6 \, {\left (a^{2} b^{2} c^{2} - a^{2} c d q\right )} x^{4} + d^{2} q^{2} - 2 \, {\left (3 \, b^{2} c d p - d^{2} p q\right )} x^{3} + 4 \, {\left (a b^{3} c^{2} - 3 \, a b c d q\right )} x^{2} - 4 \, {\left (a^{3} c x^{6} + 3 \, a^{2} b c x^{4} - a d p x^{5} - b d p x^{3} + b^{3} c - b d q + {\left (3 \, a b^{2} c - a d q\right )} x^{2}\right )} \sqrt {p x^{3} + q} \sqrt {-c d}}{a^{4} c^{2} x^{8} + 2 \, a^{2} c d p x^{7} + 4 \, a b c d p x^{5} + {\left (4 \, a^{3} b c^{2} + d^{2} p^{2}\right )} x^{6} + b^{4} c^{2} + 2 \, b^{2} c d q + 2 \, {\left (3 \, a^{2} b^{2} c^{2} + a^{2} c d q\right )} x^{4} + d^{2} q^{2} + 2 \, {\left (b^{2} c d p + d^{2} p q\right )} x^{3} + 4 \, {\left (a b^{3} c^{2} + a b c d q\right )} x^{2}}\right )}{2 \, c d}, \frac {\sqrt {c d} \arctan \left (\frac {{\left (a^{2} c x^{4} + 2 \, a b c x^{2} - d p x^{3} + b^{2} c - d q\right )} \sqrt {p x^{3} + q} \sqrt {c d}}{2 \, {\left (a c d p x^{5} + b c d p x^{3} + a c d q x^{2} + b c d q\right )}}\right )}{c d}\right ] \]
integrate((a*p*x^4-3*b*p*x^2+4*a*q*x)/(p*x^3+q)^(1/2)/(a^2*c*x^4+2*a*b*c*x ^2+d*p*x^3+b^2*c+d*q),x, algorithm="fricas")
[-1/2*sqrt(-c*d)*log((a^4*c^2*x^8 - 6*a^2*c*d*p*x^7 - 12*a*b*c*d*p*x^5 + ( 4*a^3*b*c^2 + d^2*p^2)*x^6 + b^4*c^2 - 6*b^2*c*d*q + 6*(a^2*b^2*c^2 - a^2* c*d*q)*x^4 + d^2*q^2 - 2*(3*b^2*c*d*p - d^2*p*q)*x^3 + 4*(a*b^3*c^2 - 3*a* b*c*d*q)*x^2 - 4*(a^3*c*x^6 + 3*a^2*b*c*x^4 - a*d*p*x^5 - b*d*p*x^3 + b^3* c - b*d*q + (3*a*b^2*c - a*d*q)*x^2)*sqrt(p*x^3 + q)*sqrt(-c*d))/(a^4*c^2* x^8 + 2*a^2*c*d*p*x^7 + 4*a*b*c*d*p*x^5 + (4*a^3*b*c^2 + d^2*p^2)*x^6 + b^ 4*c^2 + 2*b^2*c*d*q + 2*(3*a^2*b^2*c^2 + a^2*c*d*q)*x^4 + d^2*q^2 + 2*(b^2 *c*d*p + d^2*p*q)*x^3 + 4*(a*b^3*c^2 + a*b*c*d*q)*x^2))/(c*d), sqrt(c*d)*a rctan(1/2*(a^2*c*x^4 + 2*a*b*c*x^2 - d*p*x^3 + b^2*c - d*q)*sqrt(p*x^3 + q )*sqrt(c*d)/(a*c*d*p*x^5 + b*c*d*p*x^3 + a*c*d*q*x^2 + b*c*d*q))/(c*d)]
Timed out. \[ \int \frac {4 a q x-3 b p x^2+a p x^4}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right )} \, dx=\text {Timed out} \]
integrate((a*p*x**4-3*b*p*x**2+4*a*q*x)/(p*x**3+q)**(1/2)/(a**2*c*x**4+2*a *b*c*x**2+d*p*x**3+b**2*c+d*q),x)
\[ \int \frac {4 a q x-3 b p x^2+a p x^4}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right )} \, dx=\int { \frac {a p x^{4} - 3 \, b p x^{2} + 4 \, a q x}{{\left (a^{2} c x^{4} + 2 \, a b c x^{2} + d p x^{3} + b^{2} c + d q\right )} \sqrt {p x^{3} + q}} \,d x } \]
integrate((a*p*x^4-3*b*p*x^2+4*a*q*x)/(p*x^3+q)^(1/2)/(a^2*c*x^4+2*a*b*c*x ^2+d*p*x^3+b^2*c+d*q),x, algorithm="maxima")
integrate((a*p*x^4 - 3*b*p*x^2 + 4*a*q*x)/((a^2*c*x^4 + 2*a*b*c*x^2 + d*p* x^3 + b^2*c + d*q)*sqrt(p*x^3 + q)), x)
Timed out. \[ \int \frac {4 a q x-3 b p x^2+a p x^4}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right )} \, dx=\text {Timed out} \]
integrate((a*p*x^4-3*b*p*x^2+4*a*q*x)/(p*x^3+q)^(1/2)/(a^2*c*x^4+2*a*b*c*x ^2+d*p*x^3+b^2*c+d*q),x, algorithm="giac")
Time = 101.05 (sec) , antiderivative size = 1058, normalized size of antiderivative = 24.05 \[ \int \frac {4 a q x-3 b p x^2+a p x^4}{\sqrt {q+p x^3} \left (b^2 c+d q+2 a b c x^2+d p x^3+a^2 c x^4\right )} \, dx=\frac {\ln \left (\frac {\left (-2\,a\,\sqrt {c}\,\sqrt {d}\,\sqrt {p\,x^3+q}+a^2\,c\,x^2\,1{}\mathrm {i}+a\,b\,c\,1{}\mathrm {i}-d\,p\,x\,1{}\mathrm {i}\right )\,\left (a^4\,b^2\,c^2\,q\,1{}\mathrm {i}+a^6\,c^2\,q\,x^4\,1{}\mathrm {i}-b^2\,d^2\,p^3\,x\,1{}\mathrm {i}+a^4\,c\,d\,q^2\,4{}\mathrm {i}+a\,b^3\,c\,d\,p^2\,1{}\mathrm {i}+a^3\,b^3\,c^2\,p\,x\,1{}\mathrm {i}+a^5\,b\,c^2\,p\,x^5\,1{}\mathrm {i}+a\,b\,d^2\,p^3\,x^3\,1{}\mathrm {i}+a^5\,b\,c^2\,q\,x^2\,2{}\mathrm {i}+a^4\,b^2\,c^2\,p\,x^3\,2{}\mathrm {i}+a^2\,d^2\,p^2\,q\,x^2\,1{}\mathrm {i}-a^2\,b^2\,c\,d\,p^2\,x^2\,1{}\mathrm {i}+a^4\,c\,d\,p\,q\,x^3\,2{}\mathrm {i}+2\,a\,b^2\,\sqrt {c}\,d^{3/2}\,p^2\,\sqrt {p\,x^3+q}+a^3\,b\,c\,d\,p^2\,x^4\,2{}\mathrm {i}+a^3\,b\,c\,d\,p\,q\,x\,2{}\mathrm {i}\right )\,\left (-a^3\,b^3\,c^2\,q\,2{}\mathrm {i}+a^3\,b^3\,c^2\,\left (p\,x^3+q\right )\,3{}\mathrm {i}+a^6\,c^2\,q\,x^6\,1{}\mathrm {i}-b^2\,d^2\,p^2\,\left (p\,x^3+q\right )\,1{}\mathrm {i}+b^4\,c\,d\,p^2\,1{}\mathrm {i}+a^3\,b\,c\,d\,q^2\,1{}\mathrm {i}+a^3\,b\,c\,d\,{\left (p\,x^3+q\right )}^2\,2{}\mathrm {i}-2\,a^3\,\sqrt {c}\,d^{3/2}\,q^2\,\sqrt {p\,x^3+q}-2\,b^3\,\sqrt {c}\,d^{3/2}\,p^2\,\sqrt {p\,x^3+q}+a^2\,b^4\,c^2\,p\,x\,1{}\mathrm {i}+a^5\,b\,c^2\,q\,x^4\,2{}\mathrm {i}+a^4\,c\,d\,q^2\,x^2\,1{}\mathrm {i}+a^5\,b\,c^2\,x^4\,\left (p\,x^3+q\right )\,1{}\mathrm {i}+a^4\,b^2\,c^2\,x^2\,\left (p\,x^3+q\right )\,3{}\mathrm {i}+a\,b\,d^2\,p^2\,x^2\,\left (p\,x^3+q\right )\,1{}\mathrm {i}+a^4\,c\,d\,q\,x^2\,\left (p\,x^3+q\right )\,2{}\mathrm {i}+a^2\,d^2\,p\,q\,x\,\left (p\,x^3+q\right )\,1{}\mathrm {i}+a\,b^3\,c\,d\,p^2\,x^2\,1{}\mathrm {i}+a^2\,b^2\,c\,d\,p\,x\,\left (p\,x^3+q\right )\,2{}\mathrm {i}\right )}{\left (c\,a^2\,x^4+2\,c\,a\,b\,x^2+c\,b^2+d\,p\,x^3+d\,q\right )\,\left (a^4\,c^2\,x^4+2\,a^3\,b\,c^2\,x^2+a^2\,b^2\,c^2+2\,a^2\,c\,d\,p\,x^3+4\,q\,a^2\,c\,d-2\,a\,b\,c\,d\,p\,x+d^2\,p^2\,x^2\right )\,\left (a^8\,c^2\,q^2\,x^4+2\,a^7\,b\,c^2\,p\,q\,x^5+2\,a^7\,b\,c^2\,q^2\,x^2+a^6\,b^2\,c^2\,p^2\,x^6+4\,a^6\,b^2\,c^2\,p\,q\,x^3+a^6\,b^2\,c^2\,q^2+2\,a^6\,c\,d\,p\,q^2\,x^3+4\,a^6\,c\,d\,q^3+2\,a^5\,b^3\,c^2\,p^2\,x^4+2\,a^5\,b^3\,c^2\,p\,q\,x+4\,a^5\,b\,c\,d\,p^2\,q\,x^4+6\,a^5\,b\,c\,d\,p\,q^2\,x+a^4\,b^4\,c^2\,p^2\,x^2+2\,a^4\,b^2\,c\,d\,p^3\,x^5+2\,a^4\,b^2\,c\,d\,p^2\,q\,x^2+a^4\,d^2\,p^2\,q^2\,x^2+2\,a^3\,b^3\,c\,d\,p^2\,q+2\,a^3\,b\,d^2\,p^3\,q\,x^3+2\,a^2\,b^4\,c\,d\,p^3\,x+a^2\,b^2\,d^2\,p^4\,x^4-2\,a^2\,b^2\,d^2\,p^3\,q\,x-2\,a\,b^3\,d^2\,p^4\,x^2+b^4\,d^2\,p^4\right )}\right )\,1{}\mathrm {i}}{\sqrt {c}\,\sqrt {d}} \]
int((a*p*x^4 - 3*b*p*x^2 + 4*a*q*x)/((q + p*x^3)^(1/2)*(d*q + b^2*c + d*p* x^3 + a^2*c*x^4 + 2*a*b*c*x^2)),x)
(log(((a^2*c*x^2*1i + a*b*c*1i - d*p*x*1i - 2*a*c^(1/2)*d^(1/2)*(q + p*x^3 )^(1/2))*(a^4*b^2*c^2*q*1i + a^6*c^2*q*x^4*1i - b^2*d^2*p^3*x*1i + a^4*c*d *q^2*4i + a*b^3*c*d*p^2*1i + a^3*b^3*c^2*p*x*1i + a^5*b*c^2*p*x^5*1i + a*b *d^2*p^3*x^3*1i + a^5*b*c^2*q*x^2*2i + a^4*b^2*c^2*p*x^3*2i + a^2*d^2*p^2* q*x^2*1i - a^2*b^2*c*d*p^2*x^2*1i + a^4*c*d*p*q*x^3*2i + 2*a*b^2*c^(1/2)*d ^(3/2)*p^2*(q + p*x^3)^(1/2) + a^3*b*c*d*p^2*x^4*2i + a^3*b*c*d*p*q*x*2i)* (a^3*b^3*c^2*(q + p*x^3)*3i - a^3*b^3*c^2*q*2i + a^6*c^2*q*x^6*1i - b^2*d^ 2*p^2*(q + p*x^3)*1i + b^4*c*d*p^2*1i + a^3*b*c*d*q^2*1i + a^3*b*c*d*(q + p*x^3)^2*2i - 2*a^3*c^(1/2)*d^(3/2)*q^2*(q + p*x^3)^(1/2) - 2*b^3*c^(1/2)* d^(3/2)*p^2*(q + p*x^3)^(1/2) + a^2*b^4*c^2*p*x*1i + a^5*b*c^2*q*x^4*2i + a^4*c*d*q^2*x^2*1i + a^5*b*c^2*x^4*(q + p*x^3)*1i + a^4*b^2*c^2*x^2*(q + p *x^3)*3i + a*b*d^2*p^2*x^2*(q + p*x^3)*1i + a^4*c*d*q*x^2*(q + p*x^3)*2i + a^2*d^2*p*q*x*(q + p*x^3)*1i + a*b^3*c*d*p^2*x^2*1i + a^2*b^2*c*d*p*x*(q + p*x^3)*2i))/((d*q + b^2*c + d*p*x^3 + a^2*c*x^4 + 2*a*b*c*x^2)*(a^2*b^2* c^2 + a^4*c^2*x^4 + d^2*p^2*x^2 + 2*a^3*b*c^2*x^2 + 4*a^2*c*d*q + 2*a^2*c* d*p*x^3 - 2*a*b*c*d*p*x)*(b^4*d^2*p^4 + a^6*b^2*c^2*q^2 + a^8*c^2*q^2*x^4 + 4*a^6*c*d*q^3 + a^4*b^4*c^2*p^2*x^2 + 2*a^5*b^3*c^2*p^2*x^4 + a^6*b^2*c^ 2*p^2*x^6 + a^2*b^2*d^2*p^4*x^4 + a^4*d^2*p^2*q^2*x^2 - 2*a*b^3*d^2*p^4*x^ 2 + 2*a^7*b*c^2*q^2*x^2 + 2*a^4*b^2*c*d*p^3*x^5 + 4*a^6*b^2*c^2*p*q*x^3 - 2*a^2*b^2*d^2*p^3*q*x + 2*a^3*b*d^2*p^3*q*x^3 + 2*a^3*b^3*c*d*p^2*q + 2...