\(\int \frac {(b+a x)^2 (-2 a q+3 b p x^2+a p x^3)}{\sqrt {q+p x^3} (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+(4 a^3 b c+2 d p q) x^3+a^4 c x^4+d p^2 x^6)} \, dx\) [2782]

Optimal result

Integrand size = 103, antiderivative size = 267 \[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\frac {\arctan \left (\frac {b \sqrt [4]{c}+a \sqrt [4]{c} x}{b \sqrt [4]{c}+a \sqrt [4]{c} x-\sqrt {2} \sqrt [4]{d} \sqrt {q+p x^3}}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}}-\frac {\arctan \left (\frac {b \sqrt [4]{c}+a \sqrt [4]{c} x}{b \sqrt [4]{c}+a \sqrt [4]{c} x+\sqrt {2} \sqrt [4]{d} \sqrt {q+p x^3}}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt {2} b \sqrt [4]{c} \sqrt [4]{d}+\sqrt {2} a \sqrt [4]{c} \sqrt [4]{d} x\right ) \sqrt {q+p x^3}}{b^2 \sqrt {c}+\sqrt {d} q+2 a b \sqrt {c} x+a^2 \sqrt {c} x^2+\sqrt {d} p x^3}\right )}{\sqrt {2} c^{3/4} \sqrt [4]{d}} \]

[Out]

1/2*arctan((b*c^(1/4)+a*c^(1/4)*x)/(b*c^(1/4)+a*c^(1/4)*x-2^(1/2)*d^(1/4)*(p*x^3+q)^(1/2)))*2^(1/2)/c^(3/4)/d^
(1/4)-1/2*arctan((b*c^(1/4)+a*c^(1/4)*x)/(b*c^(1/4)+a*c^(1/4)*x+2^(1/2)*d^(1/4)*(p*x^3+q)^(1/2)))*2^(1/2)/c^(3
/4)/d^(1/4)+1/2*arctanh((2^(1/2)*b*c^(1/4)*d^(1/4)+2^(1/2)*a*c^(1/4)*d^(1/4)*x)*(p*x^3+q)^(1/2)/(b^2*c^(1/2)+d
^(1/2)*q+2*a*b*c^(1/2)*x+a^2*c^(1/2)*x^2+d^(1/2)*p*x^3))*2^(1/2)/c^(3/4)/d^(1/4)

Rubi [F]

\[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx \]

[In]

Int[((b + a*x)^2*(-2*a*q + 3*b*p*x^2 + a*p*x^3))/(Sqrt[q + p*x^3]*(b^4*c + d*q^2 + 4*a*b^3*c*x + 6*a^2*b^2*c*x
^2 + (4*a^3*b*c + 2*d*p*q)*x^3 + a^4*c*x^4 + d*p^2*x^6)),x]

[Out]

2*a*b^2*q*Defer[Int][1/(Sqrt[q + p*x^3]*(-(b^4*c) - 4*a*b^3*c*x - 6*a^2*b^2*c*x^2 - 4*a^3*b*c*x^3 - a^4*c*x^4
- d*(q + p*x^3)^2)), x] + 4*a^2*b*q*Defer[Int][x/(Sqrt[q + p*x^3]*(-(b^4*c) - 4*a*b^3*c*x - 6*a^2*b^2*c*x^2 -
4*a^3*b*c*x^3 - a^4*c*x^4 - d*(q + p*x^3)^2)), x] - (3*b^3*p - 2*a^3*q)*Defer[Int][x^2/(Sqrt[q + p*x^3]*(-(b^4
*c) - 4*a*b^3*c*x - 6*a^2*b^2*c*x^2 - 4*a^3*b*c*x^3 - a^4*c*x^4 - d*(q + p*x^3)^2)), x] + 7*a*b^2*p*Defer[Int]
[x^3/(Sqrt[q + p*x^3]*(b^4*c + 4*a*b^3*c*x + 6*a^2*b^2*c*x^2 + 4*a^3*b*c*x^3 + a^4*c*x^4 + d*(q + p*x^3)^2)),
x] + 5*a^2*b*p*Defer[Int][x^4/(Sqrt[q + p*x^3]*(b^4*c + 4*a*b^3*c*x + 6*a^2*b^2*c*x^2 + 4*a^3*b*c*x^3 + a^4*c*
x^4 + d*(q + p*x^3)^2)), x] + a^3*p*Defer[Int][x^5/(Sqrt[q + p*x^3]*(b^4*c + 4*a*b^3*c*x + 6*a^2*b^2*c*x^2 + 4
*a^3*b*c*x^3 + a^4*c*x^4 + d*(q + p*x^3)^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx \\ & = \int \left (\frac {2 a b^2 q}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )}+\frac {4 a^2 b q x}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )}+\frac {2 a^3 \left (1-\frac {3 b^3 p}{2 a^3 q}\right ) q x^2}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )}+\frac {7 a b^2 p x^3}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )}+\frac {5 a^2 b p x^4}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )}+\frac {a^3 p x^5}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )}\right ) \, dx \\ & = \left (a^3 p\right ) \int \frac {x^5}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx+\left (5 a^2 b p\right ) \int \frac {x^4}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx+\left (7 a b^2 p\right ) \int \frac {x^3}{\sqrt {q+p x^3} \left (b^4 c \left (1+\frac {d q^2}{b^4 c}\right )+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx+\left (4 a^2 b q\right ) \int \frac {x}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )} \, dx+\left (2 a b^2 q\right ) \int \frac {1}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )} \, dx+\left (-3 b^3 p+2 a^3 q\right ) \int \frac {x^2}{\sqrt {q+p x^3} \left (-b^4 c \left (1+\frac {d q^2}{b^4 c}\right )-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c \left (1+\frac {d p q}{2 a^3 b c}\right ) x^3-a^4 c x^4-d p^2 x^6\right )} \, dx \\ & = \left (a^3 p\right ) \int \frac {x^5}{\sqrt {q+p x^3} \left (b^4 c+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c x^3+a^4 c x^4+d \left (q+p x^3\right )^2\right )} \, dx+\left (5 a^2 b p\right ) \int \frac {x^4}{\sqrt {q+p x^3} \left (b^4 c+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c x^3+a^4 c x^4+d \left (q+p x^3\right )^2\right )} \, dx+\left (7 a b^2 p\right ) \int \frac {x^3}{\sqrt {q+p x^3} \left (b^4 c+4 a b^3 c x+6 a^2 b^2 c x^2+4 a^3 b c x^3+a^4 c x^4+d \left (q+p x^3\right )^2\right )} \, dx+\left (4 a^2 b q\right ) \int \frac {x}{\sqrt {q+p x^3} \left (-b^4 c-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c x^3-a^4 c x^4-d \left (q+p x^3\right )^2\right )} \, dx+\left (2 a b^2 q\right ) \int \frac {1}{\sqrt {q+p x^3} \left (-b^4 c-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c x^3-a^4 c x^4-d \left (q+p x^3\right )^2\right )} \, dx+\left (-3 b^3 p+2 a^3 q\right ) \int \frac {x^2}{\sqrt {q+p x^3} \left (-b^4 c-4 a b^3 c x-6 a^2 b^2 c x^2-4 a^3 b c x^3-a^4 c x^4-d \left (q+p x^3\right )^2\right )} \, dx \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 16.87 (sec) , antiderivative size = 52633, normalized size of antiderivative = 197.13 \[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\text {Result too large to show} \]

[In]

Integrate[((b + a*x)^2*(-2*a*q + 3*b*p*x^2 + a*p*x^3))/(Sqrt[q + p*x^3]*(b^4*c + d*q^2 + 4*a*b^3*c*x + 6*a^2*b
^2*c*x^2 + (4*a^3*b*c + 2*d*p*q)*x^3 + a^4*c*x^4 + d*p^2*x^6)),x]

[Out]

Result too large to show

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.63 (sec) , antiderivative size = 7274, normalized size of antiderivative = 27.24

method	result	size
default	\(\text {Expression too large to display}\)	\(7274\)
elliptic	\(\text {Expression too large to display}\)	\(7274\)

[In]

int((a*x+b)^2*(a*p*x^3+3*b*p*x^2-2*a*q)/(p*x^3+q)^(1/2)/(b^4*c+d*q^2+4*a*b^3*c*x+6*a^2*b^2*c*x^2+(4*a^3*b*c+2*
d*p*q)*x^3+a^4*c*x^4+d*p^2*x^6),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*x+b)^2*(a*p*x^3+3*b*p*x^2-2*a*q)/(p*x^3+q)^(1/2)/(b^4*c+d*q^2+4*a*b^3*c*x+6*a^2*b^2*c*x^2+(4*a^3*
b*c+2*d*p*q)*x^3+a^4*c*x^4+d*p^2*x^6),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\int \frac {\left (a x + b\right )^{2} \left (a p x^{3} - 2 a q + 3 b p x^{2}\right )}{\sqrt {p x^{3} + q} \left (a^{4} c x^{4} + 4 a^{3} b c x^{3} + 6 a^{2} b^{2} c x^{2} + 4 a b^{3} c x + b^{4} c + d p^{2} x^{6} + 2 d p q x^{3} + d q^{2}\right )}\, dx \]

[In]

integrate((a*x+b)**2*(a*p*x**3+3*b*p*x**2-2*a*q)/(p*x**3+q)**(1/2)/(b**4*c+d*q**2+4*a*b**3*c*x+6*a**2*b**2*c*x
**2+(4*a**3*b*c+2*d*p*q)*x**3+a**4*c*x**4+d*p**2*x**6),x)

[Out]

Integral((a*x + b)**2*(a*p*x**3 - 2*a*q + 3*b*p*x**2)/(sqrt(p*x**3 + q)*(a**4*c*x**4 + 4*a**3*b*c*x**3 + 6*a**
2*b**2*c*x**2 + 4*a*b**3*c*x + b**4*c + d*p**2*x**6 + 2*d*p*q*x**3 + d*q**2)), x)

Maxima [F]

\[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\int { \frac {{\left (a p x^{3} + 3 \, b p x^{2} - 2 \, a q\right )} {\left (a x + b\right )}^{2}}{{\left (a^{4} c x^{4} + d p^{2} x^{6} + 6 \, a^{2} b^{2} c x^{2} + 4 \, a b^{3} c x + b^{4} c + 2 \, {\left (2 \, a^{3} b c + d p q\right )} x^{3} + d q^{2}\right )} \sqrt {p x^{3} + q}} \,d x } \]

[In]

integrate((a*x+b)^2*(a*p*x^3+3*b*p*x^2-2*a*q)/(p*x^3+q)^(1/2)/(b^4*c+d*q^2+4*a*b^3*c*x+6*a^2*b^2*c*x^2+(4*a^3*
b*c+2*d*p*q)*x^3+a^4*c*x^4+d*p^2*x^6),x, algorithm="maxima")

[Out]

integrate((a*p*x^3 + 3*b*p*x^2 - 2*a*q)*(a*x + b)^2/((a^4*c*x^4 + d*p^2*x^6 + 6*a^2*b^2*c*x^2 + 4*a*b^3*c*x +
b^4*c + 2*(2*a^3*b*c + d*p*q)*x^3 + d*q^2)*sqrt(p*x^3 + q)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*x+b)^2*(a*p*x^3+3*b*p*x^2-2*a*q)/(p*x^3+q)^(1/2)/(b^4*c+d*q^2+4*a*b^3*c*x+6*a^2*b^2*c*x^2+(4*a^3*
b*c+2*d*p*q)*x^3+a^4*c*x^4+d*p^2*x^6),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(b+a x)^2 \left (-2 a q+3 b p x^2+a p x^3\right )}{\sqrt {q+p x^3} \left (b^4 c+d q^2+4 a b^3 c x+6 a^2 b^2 c x^2+\left (4 a^3 b c+2 d p q\right ) x^3+a^4 c x^4+d p^2 x^6\right )} \, dx=\text {Hanged} \]

[In]

int(((b + a*x)^2*(a*p*x^3 - 2*a*q + 3*b*p*x^2))/((q + p*x^3)^(1/2)*(x^3*(2*d*p*q + 4*a^3*b*c) + b^4*c + d*q^2
+ a^4*c*x^4 + d*p^2*x^6 + 6*a^2*b^2*c*x^2 + 4*a*b^3*c*x)),x)

[Out]

\text{Hanged}