\(\int \frac {(d+e x)^3 (f+g x)^5}{(d^2-e^2 x^2)^{7/2}} \, dx\) [579]

Optimal result

Integrand size = 31, antiderivative size = 269 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \]

[Out]

1/5*(d*g+e*f)^5*(e*x+d)^3/d/e^6/(-e^2*x^2+d^2)^(5/2)+1/15*(-23*d*g+2*e*f)*(d*g+e*f)^4*(e*x+d)^2/d^2/e^6/(-e^2*
x^2+d^2)^(3/2)-1/2*g^3*(13*d^2*g^2+30*d*e*f*g+20*e^2*f^2)*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^6+1/15*(d*g+e*f)^
3*(127*d^2*g^2-21*d*e*f*g+2*e^2*f^2)*(e*x+d)/d^3/e^6/(-e^2*x^2+d^2)^(1/2)+g^4*(3*d*g+5*e*f)*(-e^2*x^2+d^2)^(1/
2)/e^6+1/2*g^5*x*(-e^2*x^2+d^2)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1649, 1829, 655, 223, 209} \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {g^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right )}{2 e^6}+\frac {g^4 \sqrt {d^2-e^2 x^2} (3 d g+5 e f)}{e^6}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}} \]

[In]

Int[((d + e*x)^3*(f + g*x)^5)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((e*f + d*g)^5*(d + e*x)^3)/(5*d*e^6*(d^2 - e^2*x^2)^(5/2)) + ((2*e*f - 23*d*g)*(e*f + d*g)^4*(d + e*x)^2)/(15
*d^2*e^6*(d^2 - e^2*x^2)^(3/2)) + ((e*f + d*g)^3*(2*e^2*f^2 - 21*d*e*f*g + 127*d^2*g^2)*(d + e*x))/(15*d^3*e^6
*Sqrt[d^2 - e^2*x^2]) + (g^4*(5*e*f + 3*d*g)*Sqrt[d^2 - e^2*x^2])/e^6 + (g^5*x*Sqrt[d^2 - e^2*x^2])/(2*e^5) -
(g^3*(20*e^2*f^2 + 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e^6)

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 655

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

Rule 1829

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*(q + 2*p + 1))), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-\frac {2 e^5 f^5-15 d e^4 f^4 g-30 d^2 e^3 f^3 g^2-30 d^3 e^2 f^2 g^3-15 d^4 e f g^4-3 d^5 g^5}{e^5}+\frac {5 d g^2 \left (10 e^3 f^3+10 d e^2 f^2 g+5 d^2 e f g^2+d^3 g^3\right ) x}{e^4}+\frac {5 d g^3 \left (10 e^2 f^2+5 d e f g+d^2 g^2\right ) x^2}{e^3}+\frac {5 d g^4 (5 e f+d g) x^3}{e^2}+\frac {5 d g^5 x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d} \\ & = \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d+e x) \left (\frac {2 e^5 f^5-15 d e^4 f^4 g+70 d^2 e^3 f^3 g^2+170 d^3 e^2 f^2 g^3+135 d^4 e f g^4+37 d^5 g^5}{e^5}+\frac {15 d^2 g^3 \left (10 e^2 f^2+10 d e f g+3 d^2 g^2\right ) x}{e^4}+\frac {15 d^2 g^4 (5 e f+2 d g) x^2}{e^3}+\frac {15 d^2 g^5 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2} \\ & = \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {\frac {15 d^3 g^3 \left (10 e^2 f^2+15 d e f g+6 d^2 g^2\right )}{e^5}+\frac {15 d^3 g^4 (5 e f+3 d g) x}{e^4}+\frac {15 d^3 g^5 x^2}{e^3}}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3} \\ & = \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {\int \frac {-\frac {15 d^3 g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right )}{e^3}-\frac {30 d^3 g^4 (5 e f+3 d g) x}{e^2}}{\sqrt {d^2-e^2 x^2}} \, dx}{30 d^3 e^2} \\ & = \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {\left (g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right )\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^5} \\ & = \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {\left (g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5} \\ & = \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (304 d^7 g^5+4 e^7 f^5 x^2+3 d^6 e g^4 (240 f-239 g x)-6 d e^6 f^4 x (2 f+5 g x)+2 d^2 e^5 f^3 \left (7 f^2+45 f g x+70 g^2 x^2\right )+d^5 e^2 g^3 \left (440 f^2-1710 f g x+479 g^2 x^2\right )+5 d^4 e^3 g^2 \left (8 f^3-204 f^2 g x+234 f g^2 x^2-9 g^3 x^3\right )-5 d^3 e^4 g \left (6 f^4+24 f^3 g x-128 f^2 g^2 x^2+30 f g^3 x^3+3 g^4 x^4\right )\right )}{d^3 (d-e x)^3}+30 g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{30 e^6} \]

[In]

Integrate[((d + e*x)^3*(f + g*x)^5)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(304*d^7*g^5 + 4*e^7*f^5*x^2 + 3*d^6*e*g^4*(240*f - 239*g*x) - 6*d*e^6*f^4*x*(2*f + 5*g*
x) + 2*d^2*e^5*f^3*(7*f^2 + 45*f*g*x + 70*g^2*x^2) + d^5*e^2*g^3*(440*f^2 - 1710*f*g*x + 479*g^2*x^2) + 5*d^4*
e^3*g^2*(8*f^3 - 204*f^2*g*x + 234*f*g^2*x^2 - 9*g^3*x^3) - 5*d^3*e^4*g*(6*f^4 + 24*f^3*g*x - 128*f^2*g^2*x^2
+ 30*f*g^3*x^3 + 3*g^4*x^4)))/(d^3*(d - e*x)^3) + 30*g^3*(20*e^2*f^2 + 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(e*x)/(
Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(30*e^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(247)=494\).

Time = 1.14 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.25

method	result	size
risch	\(\frac {g^{4} \left (e g x +6 d g +10 e f \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{6}}-\frac {\frac {13 d^{2} g^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {20 e^{2} f^{2} g^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {30 d e f \,g^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {20 g^{2} \left (d^{3} g^{3}+3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{e^{2} d \left (x -\frac {d}{e}\right )}+\frac {10 g \left (g^{4} d^{4}+4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}+4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{e^{2}}+\frac {\left (2 g^{5} d^{5}+10 e f \,g^{4} d^{4}+20 e^{2} f^{2} g^{3} d^{3}+20 e^{3} f^{3} g^{2} d^{2}+10 f^{4} g \,e^{4} d +2 f^{5} e^{5}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{5 d e \left (x -\frac {d}{e}\right )^{3}}-\frac {2 e \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{5 d}\right )}{e^{3}}}{2 e^{5}}\)	\(604\)
default	\(\text {Expression too large to display}\)	\(1029\)

[In]

int((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/2*g^4*(e*g*x+6*d*g+10*e*f)/e^6*(-e^2*x^2+d^2)^(1/2)-1/2/e^5*(13*d^2*g^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e
^2*x^2+d^2)^(1/2))+20*e^2*f^2*g^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+30*d*e*f*g^4/(e^2)^(1
/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+20*g^2*(d^3*g^3+3*d^2*e*f*g^2+3*d*e^2*f^2*g+e^3*f^3)/e^2/d/(x-d
/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)+10*g*(d^4*g^4+4*d^3*e*f*g^3+6*d^2*e^2*f^2*g^2+4*d*e^3*f^3*g+e^4*f^4)/
e^2*(1/3/d/e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-1/3/d^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/
2))+(2*d^5*g^5+10*d^4*e*f*g^4+20*d^3*e^2*f^2*g^3+20*d^2*e^3*f^3*g^2+10*d*e^4*f^4*g+2*e^5*f^5)/e^3*(1/5/d/e/(x-
d/e)^3*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-2/5*e/d*(1/3/d/e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-1/
3/d^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (247) = 494\).

Time = 0.33 (sec) , antiderivative size = 807, normalized size of antiderivative = 3.00 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {14 \, d^{3} e^{5} f^{5} - 30 \, d^{4} e^{4} f^{4} g + 40 \, d^{5} e^{3} f^{3} g^{2} + 440 \, d^{6} e^{2} f^{2} g^{3} + 720 \, d^{7} e f g^{4} + 304 \, d^{8} g^{5} - 2 \, {\left (7 \, e^{8} f^{5} - 15 \, d e^{7} f^{4} g + 20 \, d^{2} e^{6} f^{3} g^{2} + 220 \, d^{3} e^{5} f^{2} g^{3} + 360 \, d^{4} e^{4} f g^{4} + 152 \, d^{5} e^{3} g^{5}\right )} x^{3} + 6 \, {\left (7 \, d e^{7} f^{5} - 15 \, d^{2} e^{6} f^{4} g + 20 \, d^{3} e^{5} f^{3} g^{2} + 220 \, d^{4} e^{4} f^{2} g^{3} + 360 \, d^{5} e^{3} f g^{4} + 152 \, d^{6} e^{2} g^{5}\right )} x^{2} - 6 \, {\left (7 \, d^{2} e^{6} f^{5} - 15 \, d^{3} e^{5} f^{4} g + 20 \, d^{4} e^{4} f^{3} g^{2} + 220 \, d^{5} e^{3} f^{2} g^{3} + 360 \, d^{6} e^{2} f g^{4} + 152 \, d^{7} e g^{5}\right )} x + 30 \, {\left (20 \, d^{6} e^{2} f^{2} g^{3} + 30 \, d^{7} e f g^{4} + 13 \, d^{8} g^{5} - {\left (20 \, d^{3} e^{5} f^{2} g^{3} + 30 \, d^{4} e^{4} f g^{4} + 13 \, d^{5} e^{3} g^{5}\right )} x^{3} + 3 \, {\left (20 \, d^{4} e^{4} f^{2} g^{3} + 30 \, d^{5} e^{3} f g^{4} + 13 \, d^{6} e^{2} g^{5}\right )} x^{2} - 3 \, {\left (20 \, d^{5} e^{3} f^{2} g^{3} + 30 \, d^{6} e^{2} f g^{4} + 13 \, d^{7} e g^{5}\right )} x\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, d^{3} e^{4} g^{5} x^{4} - 14 \, d^{2} e^{5} f^{5} + 30 \, d^{3} e^{4} f^{4} g - 40 \, d^{4} e^{3} f^{3} g^{2} - 440 \, d^{5} e^{2} f^{2} g^{3} - 720 \, d^{6} e f g^{4} - 304 \, d^{7} g^{5} + 15 \, {\left (10 \, d^{3} e^{4} f g^{4} + 3 \, d^{4} e^{3} g^{5}\right )} x^{3} - {\left (4 \, e^{7} f^{5} - 30 \, d e^{6} f^{4} g + 140 \, d^{2} e^{5} f^{3} g^{2} + 640 \, d^{3} e^{4} f^{2} g^{3} + 1170 \, d^{4} e^{3} f g^{4} + 479 \, d^{5} e^{2} g^{5}\right )} x^{2} + 3 \, {\left (4 \, d e^{6} f^{5} - 30 \, d^{2} e^{5} f^{4} g + 40 \, d^{3} e^{4} f^{3} g^{2} + 340 \, d^{4} e^{3} f^{2} g^{3} + 570 \, d^{5} e^{2} f g^{4} + 239 \, d^{6} e g^{5}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{3} e^{9} x^{3} - 3 \, d^{4} e^{8} x^{2} + 3 \, d^{5} e^{7} x - d^{6} e^{6}\right )}} \]

[In]

integrate((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/30*(14*d^3*e^5*f^5 - 30*d^4*e^4*f^4*g + 40*d^5*e^3*f^3*g^2 + 440*d^6*e^2*f^2*g^3 + 720*d^7*e*f*g^4 + 304*d^
8*g^5 - 2*(7*e^8*f^5 - 15*d*e^7*f^4*g + 20*d^2*e^6*f^3*g^2 + 220*d^3*e^5*f^2*g^3 + 360*d^4*e^4*f*g^4 + 152*d^5
*e^3*g^5)*x^3 + 6*(7*d*e^7*f^5 - 15*d^2*e^6*f^4*g + 20*d^3*e^5*f^3*g^2 + 220*d^4*e^4*f^2*g^3 + 360*d^5*e^3*f*g
^4 + 152*d^6*e^2*g^5)*x^2 - 6*(7*d^2*e^6*f^5 - 15*d^3*e^5*f^4*g + 20*d^4*e^4*f^3*g^2 + 220*d^5*e^3*f^2*g^3 + 3
60*d^6*e^2*f*g^4 + 152*d^7*e*g^5)*x + 30*(20*d^6*e^2*f^2*g^3 + 30*d^7*e*f*g^4 + 13*d^8*g^5 - (20*d^3*e^5*f^2*g
^3 + 30*d^4*e^4*f*g^4 + 13*d^5*e^3*g^5)*x^3 + 3*(20*d^4*e^4*f^2*g^3 + 30*d^5*e^3*f*g^4 + 13*d^6*e^2*g^5)*x^2 -
3*(20*d^5*e^3*f^2*g^3 + 30*d^6*e^2*f*g^4 + 13*d^7*e*g^5)*x)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (15*d
^3*e^4*g^5*x^4 - 14*d^2*e^5*f^5 + 30*d^3*e^4*f^4*g - 40*d^4*e^3*f^3*g^2 - 440*d^5*e^2*f^2*g^3 - 720*d^6*e*f*g^
4 - 304*d^7*g^5 + 15*(10*d^3*e^4*f*g^4 + 3*d^4*e^3*g^5)*x^3 - (4*e^7*f^5 - 30*d*e^6*f^4*g + 140*d^2*e^5*f^3*g^
2 + 640*d^3*e^4*f^2*g^3 + 1170*d^4*e^3*f*g^4 + 479*d^5*e^2*g^5)*x^2 + 3*(4*d*e^6*f^5 - 30*d^2*e^5*f^4*g + 40*d
^3*e^4*f^3*g^2 + 340*d^4*e^3*f^2*g^3 + 570*d^5*e^2*f*g^4 + 239*d^6*e*g^5)*x)*sqrt(-e^2*x^2 + d^2))/(d^3*e^9*x^
3 - 3*d^4*e^8*x^2 + 3*d^5*e^7*x - d^6*e^6)

Sympy [F]

\[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((e*x+d)**3*(g*x+f)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**3*(f + g*x)**5/(-(-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. 1603 vs. \(2 (247) = 494\).

Time = 0.29 (sec) , antiderivative size = 1603, normalized size of antiderivative = 5.96 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-1/2*e*g^5*x^7/(-e^2*x^2 + d^2)^(5/2) + 7/30*d^2*e*g^5*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)) - 7/6*d^2*g^5*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))/e + 1/5*d*f^5*x/(-e^2*x^2 + d^2)^(5/2) + 3/5*d^2*f^5/((-e^2*x^2 +
d^2)^(5/2)*e) + d^3*f^4*g/((-e^2*x^2 + d^2)^(5/2)*e^2) + 4/15*f^5*x/((-e^2*x^2 + d^2)^(3/2)*d) + 14/15*d^4*g^5
*x/((-e^2*x^2 + d^2)^(3/2)*e^5) + 1/15*(10*e^3*f^2*g^3 + 15*d*e^2*f*g^4 + 3*d^2*e*g^5)*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)) + 8/15*f^5*x/(
sqrt(-e^2*x^2 + d^2)*d^3) - 49/30*d^2*g^5*x/(sqrt(-e^2*x^2 + d^2)*e^5) - (5*e^3*f*g^4 + 3*d*e^2*g^5)*x^6/((-e^
2*x^2 + d^2)^(5/2)*e^2) - 7/2*d^2*g^5*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^5) - 1/3*(10*e^3*f^2*g^3 + 15*d
*e^2*f*g^4 + 3*d^2*e*g^5)*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))/e^2 + 6*
(5*e^3*f*g^4 + 3*d*e^2*g^5)*d^2*x^4/((-e^2*x^2 + d^2)^(5/2)*e^4) + (10*e^3*f^3*g^2 + 30*d*e^2*f^2*g^3 + 15*d^2
*e*f*g^4 + d^3*g^5)*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) + 5/2*(e^3*f^4*g + 6*d*e^2*f^3*g^2 + 6*d^2*e*f^2*g^3 + d^
3*f*g^4)*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 8*(5*e^3*f*g^4 + 3*d*e^2*g^5)*d^4*x^2/((-e^2*x^2 + d^2)^(5/2)*e^6)
- 4/3*(10*e^3*f^3*g^2 + 30*d*e^2*f^2*g^3 + 15*d^2*e*f*g^4 + d^3*g^5)*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 1
/3*(e^3*f^5 + 15*d*e^2*f^4*g + 30*d^2*e*f^3*g^2 + 10*d^3*f^2*g^3)*x^2/((-e^2*x^2 + d^2)^(5/2)*e^2) - 3/2*(e^3*
f^4*g + 6*d*e^2*f^3*g^2 + 6*d^2*e*f^2*g^3 + d^3*f*g^4)*d^2*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 1/5*(3*d*e^2*f^5 +
15*d^2*e*f^4*g + 10*d^3*f^3*g^2)*x/((-e^2*x^2 + d^2)^(5/2)*e^2) + 16/5*(5*e^3*f*g^4 + 3*d*e^2*g^5)*d^6/((-e^2
*x^2 + d^2)^(5/2)*e^8) + 8/15*(10*e^3*f^3*g^2 + 30*d*e^2*f^2*g^3 + 15*d^2*e*f*g^4 + d^3*g^5)*d^4/((-e^2*x^2 +
d^2)^(5/2)*e^6) - 2/15*(e^3*f^5 + 15*d*e^2*f^4*g + 30*d^2*e*f^3*g^2 + 10*d^3*f^2*g^3)*d^2/((-e^2*x^2 + d^2)^(5
/2)*e^4) + 4/15*(10*e^3*f^2*g^3 + 15*d*e^2*f*g^4 + 3*d^2*e*g^5)*d^2*x/((-e^2*x^2 + d^2)^(3/2)*e^6) + 1/2*(e^3*
f^4*g + 6*d*e^2*f^3*g^2 + 6*d^2*e*f^2*g^3 + d^3*f*g^4)*x/((-e^2*x^2 + d^2)^(3/2)*e^4) - 1/15*(3*d*e^2*f^5 + 15
*d^2*e*f^4*g + 10*d^3*f^3*g^2)*x/((-e^2*x^2 + d^2)^(3/2)*d^2*e^2) - 7/15*(10*e^3*f^2*g^3 + 15*d*e^2*f*g^4 + 3*
d^2*e*g^5)*x/(sqrt(-e^2*x^2 + d^2)*e^6) + (e^3*f^4*g + 6*d*e^2*f^3*g^2 + 6*d^2*e*f^2*g^3 + d^3*f*g^4)*x/(sqrt(
-e^2*x^2 + d^2)*d^2*e^4) - 2/15*(3*d*e^2*f^5 + 15*d^2*e*f^4*g + 10*d^3*f^3*g^2)*x/(sqrt(-e^2*x^2 + d^2)*d^4*e^
2) - (10*e^3*f^2*g^3 + 15*d*e^2*f*g^4 + 3*d^2*e*g^5)*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (247) = 494\).

Time = 0.32 (sec) , antiderivative size = 969, normalized size of antiderivative = 3.60 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {g^{5} x}{e^{5}} + \frac {2 \, {\left (5 \, e^{11} f g^{4} + 3 \, d e^{10} g^{5}\right )}}{e^{16}}\right )} - \frac {{\left (20 \, e^{2} f^{2} g^{3} + 30 \, d e f g^{4} + 13 \, d^{2} g^{5}\right )} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, e^{5} {\left | e \right |}} + \frac {2 \, {\left (7 \, e^{5} f^{5} - 15 \, d e^{4} f^{4} g + 20 \, d^{2} e^{3} f^{3} g^{2} + 220 \, d^{3} e^{2} f^{2} g^{3} + 285 \, d^{4} e f g^{4} + 107 \, d^{5} g^{5} - \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{3} f^{5}}{x} + \frac {75 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{2} f^{4} g}{x} - \frac {100 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e f^{3} g^{2}}{x} - \frac {950 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3} f^{2} g^{3}}{x} - \frac {1200 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} f g^{4}}{e x} - \frac {445 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{5} g^{5}}{e^{2} x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e f^{5}}{x^{2}} - \frac {75 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d f^{4} g}{x^{2}} + \frac {200 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} f^{3} g^{2}}{e x^{2}} + \frac {1450 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{3} f^{2} g^{3}}{e^{2} x^{2}} + \frac {1800 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} f g^{4}}{e^{3} x^{2}} + \frac {665 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{5} g^{5}}{e^{4} x^{2}} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} f^{5}}{e x^{3}} + \frac {75 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d f^{4} g}{e^{2} x^{3}} - \frac {750 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3} f^{2} g^{3}}{e^{4} x^{3}} - \frac {1050 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4} f g^{4}}{e^{5} x^{3}} - \frac {405 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{5} g^{5}}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} f^{5}}{e^{3} x^{4}} + \frac {150 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3} f^{2} g^{3}}{e^{6} x^{4}} + \frac {225 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{4} f g^{4}}{e^{7} x^{4}} + \frac {90 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{5} g^{5}}{e^{8} x^{4}}\right )}}{15 \, d^{3} e^{5} {\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)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

1/2*sqrt(-e^2*x^2 + d^2)*(g^5*x/e^5 + 2*(5*e^11*f*g^4 + 3*d*e^10*g^5)/e^16) - 1/2*(20*e^2*f^2*g^3 + 30*d*e*f*g
^4 + 13*d^2*g^5)*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^5*abs(e)) + 2/15*(7*e^5*f^5 - 15*d*e^4*f^4*g + 20*d^2*e^3*f^3*
g^2 + 220*d^3*e^2*f^2*g^3 + 285*d^4*e*f*g^4 + 107*d^5*g^5 - 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^3*f^5/x +
75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*e^2*f^4*g/x - 100*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*e*f^3*g^2/
x - 950*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^3*f^2*g^3/x - 1200*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^4*f*g^4
/(e*x) - 445*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^5*g^5/(e^2*x) + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e*
f^5/x^2 - 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*f^4*g/x^2 + 200*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2
*f^3*g^2/(e*x^2) + 1450*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^3*f^2*g^3/(e^2*x^2) + 1800*(d*e + sqrt(-e^2*x^
2 + d^2)*abs(e))^2*d^4*f*g^4/(e^3*x^2) + 665*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^5*g^5/(e^4*x^2) - 30*(d*e
+ sqrt(-e^2*x^2 + d^2)*abs(e))^3*f^5/(e*x^3) + 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d*f^4*g/(e^2*x^3) - 7
50*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^3*f^2*g^3/(e^4*x^3) - 1050*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^
4*f*g^4/(e^5*x^3) - 405*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^5*g^5/(e^6*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^
2)*abs(e))^4*f^5/(e^3*x^4) + 150*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^3*f^2*g^3/(e^6*x^4) + 225*(d*e + sqrt
(-e^2*x^2 + d^2)*abs(e))^4*d^4*f*g^4/(e^7*x^4) + 90*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^5*g^5/(e^8*x^4))/(
d^3*e^5*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^5\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

[In]

int(((f + g*x)^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int(((f + g*x)^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2), x)