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

Optimal. Leaf size=183 \[ \frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \]

[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]
time = 0.25, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1649, 792, 223, 209} \begin {gather*} -\frac {g^3 \text {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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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]
time = 0.86, size = 165, normalized size = 0.90 \begin {gather*} \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 )}{15 d^3 e^4 (d-e x)^3}+\frac {g^3 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^3 \sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[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)))/(15*d^3*e^4*(d - e*x)^3) + (g^3*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2
- e^2*x^2]])/(e^3*Sqrt[-e^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(687\) vs. \(2(169)=338\).
time = 0.10, size = 688, normalized size = 3.76

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 834 vs. \(2 (169) = 338\).
time = 0.51, size = 834, normalized size = 4.56 \begin {gather*} \frac {1}{15} \, {\left (\frac {15 \, x^{4} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {20 \, d^{2} x^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, d^{4} e^{\left (-6\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}\right )} g^{3} x e^{3} - \frac {1}{3} \, {\left (\frac {3 \, x^{2} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, d^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}}\right )} g^{3} x e + \frac {4 \, d^{2} g^{3} x e^{\left (-3\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - g^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} + \frac {3 \, d^{3} f^{2} g e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {7 \, g^{3} x e^{\left (-3\right )}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}}} + \frac {3 \, d^{2} f^{3} e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, {\left (d g^{3} e^{2} + f g^{2} e^{3}\right )} x^{4} e^{\left (-2\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {4 \, {\left (d g^{3} e^{2} + f g^{2} e^{3}\right )} d^{2} x^{2} e^{\left (-4\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, {\left (d g^{3} e^{2} + f g^{2} e^{3}\right )} d^{4} e^{\left (-6\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d f^{3} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, {\left (d^{2} g^{3} e + 3 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )} x^{3} e^{\left (-2\right )}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {9 \, {\left (d^{2} g^{3} e + 3 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )} d^{2} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, f^{3} x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {{\left (d^{3} g^{3} + 9 \, d^{2} f g^{2} e + 9 \, d f^{2} g e^{2} + f^{3} e^{3}\right )} x^{2} e^{\left (-2\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {2 \, {\left (d^{3} g^{3} + 9 \, d^{2} f g^{2} e + 9 \, d f^{2} g e^{2} + f^{3} e^{3}\right )} d^{2} e^{\left (-4\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, {\left (d^{2} g^{3} e + 3 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {8 \, f^{3} x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}} + \frac {3 \, {\left (d^{3} f g^{2} + 3 \, d^{2} f^{2} g e + d f^{3} e^{2}\right )} x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, {\left (d^{2} g^{3} e + 3 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )} x e^{\left (-4\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} - \frac {{\left (d^{3} f g^{2} + 3 \, d^{2} f^{2} g e + d f^{3} e^{2}\right )} x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} - \frac {2 \, {\left (d^{3} f g^{2} + 3 \, d^{2} f^{2} g e + d f^{3} e^{2}\right )} x e^{\left (-2\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (169) = 338\).
time = 2.76, size = 442, normalized size = 2.42 \begin {gather*} -\frac {22 \, d^{6} g^{3} - 7 \, f^{3} x^{3} e^{6} - 30 \, {\left (d^{3} g^{3} x^{3} e^{3} - 3 \, d^{4} g^{3} x^{2} e^{2} + 3 \, d^{5} g^{3} x e - d^{6} g^{3}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + 3 \, {\left (3 \, d f^{2} g x^{3} + 7 \, d f^{3} x^{2}\right )} e^{5} - 3 \, {\left (2 \, d^{2} f g^{2} x^{3} + 9 \, d^{2} f^{2} g x^{2} + 7 \, d^{2} f^{3} x\right )} e^{4} - {\left (22 \, d^{3} g^{3} x^{3} - 18 \, d^{3} f g^{2} x^{2} - 27 \, d^{3} f^{2} g x - 7 \, d^{3} f^{3}\right )} e^{3} + 3 \, {\left (22 \, d^{4} g^{3} x^{2} - 6 \, d^{4} f g^{2} x - 3 \, d^{4} f^{2} g\right )} e^{2} - 6 \, {\left (11 \, d^{5} g^{3} x - d^{5} f g^{2}\right )} e + {\left (22 \, d^{5} g^{3} + 2 \, f^{3} x^{2} e^{5} - 3 \, {\left (3 \, d f^{2} g x^{2} + 2 \, d f^{3} x\right )} e^{4} + {\left (21 \, d^{2} f g^{2} x^{2} + 27 \, d^{2} f^{2} g x + 7 \, d^{2} f^{3}\right )} e^{3} + {\left (32 \, d^{3} g^{3} x^{2} - 18 \, d^{3} f g^{2} x - 9 \, d^{3} f^{2} g\right )} e^{2} - 3 \, {\left (17 \, d^{4} g^{3} x - 2 \, d^{4} f g^{2}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{3} e^{7} - 3 \, d^{4} x^{2} e^{6} + 3 \, d^{5} x e^{5} - d^{6} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(22*d^6*g^3 - 7*f^3*x^3*e^6 - 30*(d^3*g^3*x^3*e^3 - 3*d^4*g^3*x^2*e^2 + 3*d^5*g^3*x*e - d^6*g^3)*arctan(
-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + 3*(3*d*f^2*g*x^3 + 7*d*f^3*x^2)*e^5 - 3*(2*d^2*f*g^2*x^3 + 9*d^2*f^2*g
*x^2 + 7*d^2*f^3*x)*e^4 - (22*d^3*g^3*x^3 - 18*d^3*f*g^2*x^2 - 27*d^3*f^2*g*x - 7*d^3*f^3)*e^3 + 3*(22*d^4*g^3
*x^2 - 6*d^4*f*g^2*x - 3*d^4*f^2*g)*e^2 - 6*(11*d^5*g^3*x - d^5*f*g^2)*e + (22*d^5*g^3 + 2*f^3*x^2*e^5 - 3*(3*
d*f^2*g*x^2 + 2*d*f^3*x)*e^4 + (21*d^2*f*g^2*x^2 + 27*d^2*f^2*g*x + 7*d^2*f^3)*e^3 + (32*d^3*g^3*x^2 - 18*d^3*
f*g^2*x - 9*d^3*f^2*g)*e^2 - 3*(17*d^4*g^3*x - 2*d^4*f*g^2)*e)*sqrt(-x^2*e^2 + d^2))/(d^3*x^3*e^7 - 3*d^4*x^2*
e^6 + 3*d^5*x*e^5 - d^6*e^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (169) = 338\).
time = 3.45, size = 537, normalized size = 2.93 \begin {gather*} -g^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\left (d\right ) - \frac {2 \, {\left (\frac {95 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} g^{3} e^{\left (-2\right )}}{x} - \frac {145 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} g^{3} e^{\left (-4\right )}}{x^{2}} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} g^{3} e^{\left (-6\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} g^{3} e^{\left (-8\right )}}{x^{4}} - 22 \, d^{3} g^{3} - 6 \, d^{2} f g^{2} e + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} f g^{2} e^{\left (-1\right )}}{x} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} f g^{2} e^{\left (-3\right )}}{x^{2}} + 9 \, d f^{2} g e^{2} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f^{2} g e^{\left (-2\right )}}{x^{2}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f^{2} g e^{\left (-4\right )}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f^{2} g}{x} - 7 \, f^{3} e^{3} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{3} e}{x} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{3} e^{\left (-1\right )}}{x^{2}} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{3} e^{\left (-3\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{3} e^{\left (-5\right )}}{x^{4}}\right )} e^{\left (-4\right )}}{15 \, d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________