\(\int \frac {(d+e x)^3 (f+g x)^3}{(d^2-e^2 x^2)^{7/2}} \, dx\) [581]

Optimal result

Integrand size = 31, antiderivative size = 183 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \]

[Out]

1/5*(d*g+e*f)^3*(e*x+d)^3/d/e^4/(-e^2*x^2+d^2)^(5/2)+1/15*(-13*d*g+2*e*f)*(d*g+e*f)^2*(e*x+d)^2/d^2/e^4/(-e^2*
x^2+d^2)^(3/2)-g^3*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4+1/15*(d*g+e*f)*(32*d^2*g^2-11*d*e*f*g+2*e^2*f^2)*(e*x+
d)/d^3/e^4/(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1649, 792, 223, 209} \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {g^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}} \]

[In]

Int[((d + e*x)^3*(f + g*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((e*f + d*g)^3*(d + e*x)^3)/(5*d*e^4*(d^2 - e^2*x^2)^(5/2)) + ((2*e*f - 13*d*g)*(e*f + d*g)^2*(d + e*x)^2)/(15
*d^2*e^4*(d^2 - e^2*x^2)^(3/2)) + ((e*f + d*g)*(2*e^2*f^2 - 11*d*e*f*g + 32*d^2*g^2)*(d + e*x))/(15*d^3*e^4*Sq
rt[d^2 - e^2*x^2]) - (g^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^4

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 792

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a*(e*f + d*g) - (
c*d*f - a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(-d)*f*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2
*a*e*(p + 1))), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)
*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p
+ 1/2, 0] && GtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-\frac {2 e^3 f^3-9 d e^2 f^2 g-9 d^2 e f g^2-3 d^3 g^3}{e^3}+\frac {5 d g^2 (3 e f+d g) x}{e^2}+\frac {5 d g^3 x^2}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d} \\ & = \frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d+e x) \left (\frac {2 e^3 f^3-9 d e^2 f^2 g+21 d^2 e f g^2+17 d^3 g^3}{e^3}+\frac {15 d^2 g^3 x}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2} \\ & = \frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^3} \\ & = \frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^3} \\ & = \frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {(e f+d g) \sqrt {d^2-e^2 x^2} \left (22 d^4 g^2+2 e^4 f^2 x^2-d e^3 f x (6 f+11 g x)-d^3 e g (16 f+51 g x)+d^2 e^2 \left (7 f^2+33 f g x+32 g^2 x^2\right )\right )}{d^3 (d-e x)^3}+30 g^3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{15 e^4} \]

[In]

Integrate[((d + e*x)^3*(f + g*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(((e*f + d*g)*Sqrt[d^2 - e^2*x^2]*(22*d^4*g^2 + 2*e^4*f^2*x^2 - d*e^3*f*x*(6*f + 11*g*x) - d^3*e*g*(16*f + 51*
g*x) + d^2*e^2*(7*f^2 + 33*f*g*x + 32*g^2*x^2)))/(d^3*(d - e*x)^3) + 30*g^3*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2
- e^2*x^2])])/(15*e^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(687\) vs. \(2(169)=338\).

Time = 0.65 (sec) , antiderivative size = 688, normalized size of antiderivative = 3.76

method	result	size
default	\(d^{3} f^{3} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+e^{3} g^{3} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )+\left (3 d \,e^{2} g^{3}+3 e^{3} f \,g^{2}\right ) \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+\frac {3 d^{3} f^{2} g +3 d^{2} e \,f^{3}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\left (3 d^{2} e \,g^{3}+9 d \,e^{2} f \,g^{2}+3 e^{3} f^{2} g \right ) \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+\left (3 d^{3} f \,g^{2}+9 d^{2} e \,f^{2} g +3 d \,e^{2} f^{3}\right ) \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\left (d^{3} g^{3}+9 d^{2} e f \,g^{2}+9 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )\)	\(688\)

[In]

int((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

d^3*f^3*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2)
))+e^3*g^3*(1/5*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1/e^2*(1/3*x^3/e^2/(-e^2*x^2+d^2)^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^
2)^(1/2)-1/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))+(3*d*e^2*g^3+3*e^3*f*g^2)*(x^4/e^2/(-
e^2*x^2+d^2)^(5/2)-4*d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/2)))+1/5*(3*d^3*
f^2*g+3*d^2*e*f^3)/e^2/(-e^2*x^2+d^2)^(5/2)+(3*d^2*e*g^3+9*d*e^2*f*g^2+3*e^3*f^2*g)*(1/2*x^3/e^2/(-e^2*x^2+d^2
)^(5/2)-3/2*d^2/e^2*(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-1/4*d^2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x
/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2)))))+(3*d^3*f*g^2+9*d^2*e*f^2*g+3*d*e^2*f^3)*(1/4*x/e^
2/(-e^2*x^2+d^2)^(5/2)-1/4*d^2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3
*x/d^4/(-e^2*x^2+d^2)^(1/2))))+(d^3*g^3+9*d^2*e*f*g^2+9*d*e^2*f^2*g+e^3*f^3)*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)
-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (169) = 338\).

Time = 0.33 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 \, d^{3} e^{3} f^{3} - 9 \, d^{4} e^{2} f^{2} g + 6 \, d^{5} e f g^{2} + 22 \, d^{6} g^{3} - {\left (7 \, e^{6} f^{3} - 9 \, d e^{5} f^{2} g + 6 \, d^{2} e^{4} f g^{2} + 22 \, d^{3} e^{3} g^{3}\right )} x^{3} + 3 \, {\left (7 \, d e^{5} f^{3} - 9 \, d^{2} e^{4} f^{2} g + 6 \, d^{3} e^{3} f g^{2} + 22 \, d^{4} e^{2} g^{3}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{4} f^{3} - 9 \, d^{3} e^{3} f^{2} g + 6 \, d^{4} e^{2} f g^{2} + 22 \, d^{5} e g^{3}\right )} x - 30 \, {\left (d^{3} e^{3} g^{3} x^{3} - 3 \, d^{4} e^{2} g^{3} x^{2} + 3 \, d^{5} e g^{3} x - d^{6} g^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (7 \, d^{2} e^{3} f^{3} - 9 \, d^{3} e^{2} f^{2} g + 6 \, d^{4} e f g^{2} + 22 \, d^{5} g^{3} + {\left (2 \, e^{5} f^{3} - 9 \, d e^{4} f^{2} g + 21 \, d^{2} e^{3} f g^{2} + 32 \, d^{3} e^{2} g^{3}\right )} x^{2} - 3 \, {\left (2 \, d e^{4} f^{3} - 9 \, d^{2} e^{3} f^{2} g + 6 \, d^{3} e^{2} f g^{2} + 17 \, d^{4} e g^{3}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{7} x^{3} - 3 \, d^{4} e^{6} x^{2} + 3 \, d^{5} e^{5} x - d^{6} e^{4}\right )}} \]

[In]

integrate((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(7*d^3*e^3*f^3 - 9*d^4*e^2*f^2*g + 6*d^5*e*f*g^2 + 22*d^6*g^3 - (7*e^6*f^3 - 9*d*e^5*f^2*g + 6*d^2*e^4*f
*g^2 + 22*d^3*e^3*g^3)*x^3 + 3*(7*d*e^5*f^3 - 9*d^2*e^4*f^2*g + 6*d^3*e^3*f*g^2 + 22*d^4*e^2*g^3)*x^2 - 3*(7*d
^2*e^4*f^3 - 9*d^3*e^3*f^2*g + 6*d^4*e^2*f*g^2 + 22*d^5*e*g^3)*x - 30*(d^3*e^3*g^3*x^3 - 3*d^4*e^2*g^3*x^2 + 3
*d^5*e*g^3*x - d^6*g^3)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (7*d^2*e^3*f^3 - 9*d^3*e^2*f^2*g + 6*d^4*e
*f*g^2 + 22*d^5*g^3 + (2*e^5*f^3 - 9*d*e^4*f^2*g + 21*d^2*e^3*f*g^2 + 32*d^3*e^2*g^3)*x^2 - 3*(2*d*e^4*f^3 - 9
*d^2*e^3*f^2*g + 6*d^3*e^2*f*g^2 + 17*d^4*e*g^3)*x)*sqrt(-e^2*x^2 + d^2))/(d^3*e^7*x^3 - 3*d^4*e^6*x^2 + 3*d^5
*e^5*x - d^6*e^4)

Sympy [F]

\[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((e*x+d)**3*(g*x+f)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**3*(f + g*x)**3/(-(-d + e*x)*(d + e*x))**(7/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (169) = 338\).

Time = 0.28 (sec) , antiderivative size = 903, normalized size of antiderivative = 4.93 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {1}{15} \, e^{3} g^{3} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {1}{3} \, e g^{3} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {d f^{3} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f^{3}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {3 \, d^{3} f^{2} g}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {4 \, f^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {4 \, d^{2} g^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} + \frac {8 \, f^{3} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} - \frac {7 \, g^{3} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3}} + \frac {3 \, {\left (e^{3} f g^{2} + d e^{2} g^{3}\right )} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {g^{3} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}} e^{3}} + \frac {3 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {4 \, {\left (e^{3} f g^{2} + d e^{2} g^{3}\right )} d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {{\left (e^{3} f^{3} + 9 \, d e^{2} f^{2} g + 9 \, d^{2} e f g^{2} + d^{3} g^{3}\right )} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {9 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {3 \, {\left (d e^{2} f^{3} + 3 \, d^{2} e f^{2} g + d^{3} f g^{2}\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {8 \, {\left (e^{3} f g^{2} + d e^{2} g^{3}\right )} d^{4}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} - \frac {2 \, {\left (e^{3} f^{3} + 9 \, d e^{2} f^{2} g + 9 \, d^{2} e f g^{2} + d^{3} g^{3}\right )} d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {3 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {{\left (d e^{2} f^{3} + 3 \, d^{2} e f^{2} g + d^{3} f g^{2}\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {3 \, {\left (e^{3} f^{2} g + 3 \, d e^{2} f g^{2} + d^{2} e g^{3}\right )} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{4}} - \frac {2 \, {\left (d e^{2} f^{3} + 3 \, d^{2} e f^{2} g + d^{3} f g^{2}\right )} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{2}} \]

[In]

integrate((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

1/15*e^3*g^3*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x
^2 + d^2)^(5/2)*e^6)) - 1/3*e*g^3*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))
+ 1/5*d*f^3*x/(-e^2*x^2 + d^2)^(5/2) + 3/5*d^2*f^3/((-e^2*x^2 + d^2)^(5/2)*e) + 3/5*d^3*f^2*g/((-e^2*x^2 + d^2
)^(5/2)*e^2) + 4/15*f^3*x/((-e^2*x^2 + d^2)^(3/2)*d) + 4/15*d^2*g^3*x/((-e^2*x^2 + d^2)^(3/2)*e^3) + 8/15*f^3*
x/(sqrt(-e^2*x^2 + d^2)*d^3) - 7/15*g^3*x/(sqrt(-e^2*x^2 + d^2)*e^3) + 3*(e^3*f*g^2 + d*e^2*g^3)*x^4/((-e^2*x^
2 + d^2)^(5/2)*e^2) - g^3*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^3) + 3/2*(e^3*f^2*g + 3*d*e^2*f*g^2 + d^2*e
*g^3)*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 4*(e^3*f*g^2 + d*e^2*g^3)*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 1/3*
(e^3*f^3 + 9*d*e^2*f^2*g + 9*d^2*e*f*g^2 + d^3*g^3)*x^2/((-e^2*x^2 + d^2)^(5/2)*e^2) - 9/10*(e^3*f^2*g + 3*d*e
^2*f*g^2 + d^2*e*g^3)*d^2*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 3/5*(d*e^2*f^3 + 3*d^2*e*f^2*g + d^3*f*g^2)*x/((-e^
2*x^2 + d^2)^(5/2)*e^2) + 8/5*(e^3*f*g^2 + d*e^2*g^3)*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6) - 2/15*(e^3*f^3 + 9*d*e
^2*f^2*g + 9*d^2*e*f*g^2 + d^3*g^3)*d^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 3/10*(e^3*f^2*g + 3*d*e^2*f*g^2 + d^2*e
*g^3)*x/((-e^2*x^2 + d^2)^(3/2)*e^4) - 1/5*(d*e^2*f^3 + 3*d^2*e*f^2*g + d^3*f*g^2)*x/((-e^2*x^2 + d^2)^(3/2)*d
^2*e^2) + 3/5*(e^3*f^2*g + 3*d*e^2*f*g^2 + d^2*e*g^3)*x/(sqrt(-e^2*x^2 + d^2)*d^2*e^4) - 2/5*(d*e^2*f^3 + 3*d^
2*e*f^2*g + d^3*f*g^2)*x/(sqrt(-e^2*x^2 + d^2)*d^4*e^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (169) = 338\).

Time = 0.31 (sec) , antiderivative size = 560, normalized size of antiderivative = 3.06 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {g^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{e^{3} {\left | e \right |}} + \frac {2 \, {\left (7 \, e^{3} f^{3} - 9 \, d e^{2} f^{2} g + 6 \, d^{2} e f g^{2} + 22 \, d^{3} g^{3} - \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e f^{3}}{x} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d f^{2} g}{x} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} f g^{2}}{e x} - \frac {95 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3} g^{3}}{e^{2} x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} f^{3}}{e x^{2}} - \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d f^{2} g}{e^{2} x^{2}} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} f g^{2}}{e^{3} x^{2}} + \frac {145 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{3} g^{3}}{e^{4} x^{2}} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} f^{3}}{e^{3} x^{3}} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d f^{2} g}{e^{4} x^{3}} - \frac {75 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3} g^{3}}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} f^{3}}{e^{5} x^{4}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3} g^{3}}{e^{8} x^{4}}\right )}}{15 \, d^{3} e^{3} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]

[In]

integrate((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-g^3*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^3*abs(e)) + 2/15*(7*e^3*f^3 - 9*d*e^2*f^2*g + 6*d^2*e*f*g^2 + 22*d^3*g^3 -
20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e*f^3/x + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*f^2*g/x - 30*(d*e +
sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*f*g^2/(e*x) - 95*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^3*g^3/(e^2*x) + 40*(d
*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*f^3/(e*x^2) - 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*f^2*g/(e^2*x^2) +
60*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2*f*g^2/(e^3*x^2) + 145*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^3*
g^3/(e^4*x^2) - 30*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*f^3/(e^3*x^3) + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)
)^3*d*f^2*g/(e^4*x^3) - 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^3*g^3/(e^6*x^3) + 15*(d*e + sqrt(-e^2*x^2 +
d^2)*abs(e))^4*f^3/(e^5*x^4) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^3*g^3/(e^8*x^4))/(d^3*e^3*((d*e + s
qrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

[In]

int(((f + g*x)^3*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int(((f + g*x)^3*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2), x)