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

Optimal. Leaf size=269 \[ \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \]

[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

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Rubi [A]
time = 0.60, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1649, 1829, 655, 223, 209} \begin {gather*} -\frac {g^3 \text {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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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*}

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Mathematica [A]
time = 1.46, size = 311, normalized size = 1.16 \begin {gather*} \frac {\frac {e^3 \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}+15 \left (-e^2\right )^{3/2} g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{30 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e^3*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) + 15*(-e^2)^(3/2)*g^3*(20*e^2*f^2 + 30*d*e*f*g + 13*d^2*g^
2)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(30*e^9)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1028\) vs. \(2(247)=494\).
time = 0.14, size = 1029, normalized size = 3.83 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^3*g^5*(-1/2*x^7/e^2/(-e^2*x^2+d^2)^(5/2)+7/2*d^2/e^2*(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^5+5*e^3*f*g^4)*(-x^6/e^2/(-e^2*x^2+d^2)^(5/2)+6*d^2/e^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^5+15*d*e^2*f*g^4+10*e^3
*f^2*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)))))+(d^3*g^5+15*d^2*e*f*g^4+30*d*e^2*f^2*g
^3+10*e^3*f^3*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)))+(5*d^3*f*g^4+30*d^2*e*f^2*g^3+30*d*e^2*f^3*g^2+5*e^3*f^4*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)))))+(10*d^3*f^2*g^3+30*d^2*e*f^3*g^2+15*d*e^2*f^4*g+e^3*f^
5)*(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))+(10*d^3*f^3*g^2+15*d^2*e*f^4*g+3*d*e^2
*f^5)*(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*(5*d^3*f^4*g+3*d^2*e*f^5)/e^2/(-e^2*x^2+d^2)^(5/2)+d^3*f^5*(
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)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1489 vs. \(2 (244) = 488\).
time = 0.52, size = 1489, normalized size = 5.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (244) = 488\).
time = 3.08, size = 781, normalized size = 2.90 \begin {gather*} -\frac {304 \, d^{8} g^{5} - 14 \, f^{5} x^{3} e^{8} + 30 \, {\left (13 \, d^{8} g^{5} - 20 \, d^{3} f^{2} g^{3} x^{3} e^{5} - 30 \, {\left (d^{4} f g^{4} x^{3} - 2 \, d^{4} f^{2} g^{3} x^{2}\right )} e^{4} - {\left (13 \, d^{5} g^{5} x^{3} - 90 \, d^{5} f g^{4} x^{2} + 60 \, d^{5} f^{2} g^{3} x\right )} e^{3} + {\left (39 \, d^{6} g^{5} x^{2} - 90 \, d^{6} f g^{4} x + 20 \, d^{6} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (13 \, d^{7} g^{5} x - 10 \, d^{7} f g^{4}\right )} e\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + 6 \, {\left (5 \, d f^{4} g x^{3} + 7 \, d f^{5} x^{2}\right )} e^{7} - 2 \, {\left (20 \, d^{2} f^{3} g^{2} x^{3} + 45 \, d^{2} f^{4} g x^{2} + 21 \, d^{2} f^{5} x\right )} e^{6} - 2 \, {\left (220 \, d^{3} f^{2} g^{3} x^{3} - 60 \, d^{3} f^{3} g^{2} x^{2} - 45 \, d^{3} f^{4} g x - 7 \, d^{3} f^{5}\right )} e^{5} - 30 \, {\left (24 \, d^{4} f g^{4} x^{3} - 44 \, d^{4} f^{2} g^{3} x^{2} + 4 \, d^{4} f^{3} g^{2} x + d^{4} f^{4} g\right )} e^{4} - 8 \, {\left (38 \, d^{5} g^{5} x^{3} - 270 \, d^{5} f g^{4} x^{2} + 165 \, d^{5} f^{2} g^{3} x - 5 \, d^{5} f^{3} g^{2}\right )} e^{3} + 8 \, {\left (114 \, d^{6} g^{5} x^{2} - 270 \, d^{6} f g^{4} x + 55 \, d^{6} f^{2} g^{3}\right )} e^{2} - 48 \, {\left (19 \, d^{7} g^{5} x - 15 \, d^{7} f g^{4}\right )} e + {\left (304 \, d^{7} g^{5} + 4 \, f^{5} x^{2} e^{7} - 6 \, {\left (5 \, d f^{4} g x^{2} + 2 \, d f^{5} x\right )} e^{6} + 2 \, {\left (70 \, d^{2} f^{3} g^{2} x^{2} + 45 \, d^{2} f^{4} g x + 7 \, d^{2} f^{5}\right )} e^{5} - 5 \, {\left (3 \, d^{3} g^{5} x^{4} + 30 \, d^{3} f g^{4} x^{3} - 128 \, d^{3} f^{2} g^{3} x^{2} + 24 \, d^{3} f^{3} g^{2} x + 6 \, d^{3} f^{4} g\right )} e^{4} - 5 \, {\left (9 \, d^{4} g^{5} x^{3} - 234 \, d^{4} f g^{4} x^{2} + 204 \, d^{4} f^{2} g^{3} x - 8 \, d^{4} f^{3} g^{2}\right )} e^{3} + {\left (479 \, d^{5} g^{5} x^{2} - 1710 \, d^{5} f g^{4} x + 440 \, d^{5} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (239 \, d^{6} g^{5} x - 240 \, d^{6} f g^{4}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (d^{3} x^{3} e^{9} - 3 \, d^{4} x^{2} e^{8} + 3 \, d^{5} x e^{7} - d^{6} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(304*d^8*g^5 - 14*f^5*x^3*e^8 + 30*(13*d^8*g^5 - 20*d^3*f^2*g^3*x^3*e^5 - 30*(d^4*f*g^4*x^3 - 2*d^4*f^2*
g^3*x^2)*e^4 - (13*d^5*g^5*x^3 - 90*d^5*f*g^4*x^2 + 60*d^5*f^2*g^3*x)*e^3 + (39*d^6*g^5*x^2 - 90*d^6*f*g^4*x +
20*d^6*f^2*g^3)*e^2 - 3*(13*d^7*g^5*x - 10*d^7*f*g^4)*e)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) + 6*(5*
d*f^4*g*x^3 + 7*d*f^5*x^2)*e^7 - 2*(20*d^2*f^3*g^2*x^3 + 45*d^2*f^4*g*x^2 + 21*d^2*f^5*x)*e^6 - 2*(220*d^3*f^2
*g^3*x^3 - 60*d^3*f^3*g^2*x^2 - 45*d^3*f^4*g*x - 7*d^3*f^5)*e^5 - 30*(24*d^4*f*g^4*x^3 - 44*d^4*f^2*g^3*x^2 +
4*d^4*f^3*g^2*x + d^4*f^4*g)*e^4 - 8*(38*d^5*g^5*x^3 - 270*d^5*f*g^4*x^2 + 165*d^5*f^2*g^3*x - 5*d^5*f^3*g^2)*
e^3 + 8*(114*d^6*g^5*x^2 - 270*d^6*f*g^4*x + 55*d^6*f^2*g^3)*e^2 - 48*(19*d^7*g^5*x - 15*d^7*f*g^4)*e + (304*d
^7*g^5 + 4*f^5*x^2*e^7 - 6*(5*d*f^4*g*x^2 + 2*d*f^5*x)*e^6 + 2*(70*d^2*f^3*g^2*x^2 + 45*d^2*f^4*g*x + 7*d^2*f^
5)*e^5 - 5*(3*d^3*g^5*x^4 + 30*d^3*f*g^4*x^3 - 128*d^3*f^2*g^3*x^2 + 24*d^3*f^3*g^2*x + 6*d^3*f^4*g)*e^4 - 5*(
9*d^4*g^5*x^3 - 234*d^4*f*g^4*x^2 + 204*d^4*f^2*g^3*x - 8*d^4*f^3*g^2)*e^3 + (479*d^5*g^5*x^2 - 1710*d^5*f*g^4
*x + 440*d^5*f^2*g^3)*e^2 - 3*(239*d^6*g^5*x - 240*d^6*f*g^4)*e)*sqrt(-x^2*e^2 + d^2))/(d^3*x^3*e^9 - 3*d^4*x^
2*e^8 + 3*d^5*x*e^7 - d^6*e^6)

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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 )^{5}}{\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)**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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (244) = 488\).
time = 1.33, size = 934, normalized size = 3.47 \begin {gather*} -\frac {1}{2} \, {\left (13 \, d^{2} g^{5} + 30 \, d f g^{4} e + 20 \, f^{2} g^{3} e^{2}\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{2} \, {\left (g^{5} x e^{\left (-5\right )} + 2 \, {\left (3 \, d g^{5} e^{10} + 5 \, f g^{4} e^{11}\right )} e^{\left (-16\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}} - \frac {2 \, {\left (\frac {445 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{5} g^{5} e^{\left (-2\right )}}{x} - \frac {665 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} g^{5} e^{\left (-4\right )}}{x^{2}} + \frac {405 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{5} g^{5} e^{\left (-6\right )}}{x^{3}} - \frac {90 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{5} g^{5} e^{\left (-8\right )}}{x^{4}} - 107 \, d^{5} g^{5} - 285 \, d^{4} f g^{4} e + \frac {1200 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} f g^{4} e^{\left (-1\right )}}{x} - \frac {1800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} f g^{4} e^{\left (-3\right )}}{x^{2}} + \frac {1050 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} f g^{4} e^{\left (-5\right )}}{x^{3}} - \frac {225 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} f g^{4} e^{\left (-7\right )}}{x^{4}} - 220 \, d^{3} f^{2} g^{3} e^{2} - \frac {1450 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} f^{2} g^{3} e^{\left (-2\right )}}{x^{2}} + \frac {750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} f^{2} g^{3} e^{\left (-4\right )}}{x^{3}} - \frac {150 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} f^{2} g^{3} e^{\left (-6\right )}}{x^{4}} + \frac {950 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} f^{2} g^{3}}{x} - 20 \, d^{2} f^{3} g^{2} e^{3} + \frac {100 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} f^{3} g^{2} e}{x} - \frac {200 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} f^{3} g^{2} e^{\left (-1\right )}}{x^{2}} + 15 \, d f^{4} g e^{4} - \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f^{4} g e^{2}}{x} - \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f^{4} g e^{\left (-2\right )}}{x^{3}} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f^{4} g}{x^{2}} - 7 \, f^{5} e^{5} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{5} e^{3}}{x} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{5} e}{x^{2}} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{5} e^{\left (-1\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{5} e^{\left (-3\right )}}{x^{4}}\right )} e^{\left (-6\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)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-1/2*(13*d^2*g^5 + 30*d*f*g^4*e + 20*f^2*g^3*e^2)*arcsin(x*e/d)*e^(-6)*sgn(d) + 1/2*(g^5*x*e^(-5) + 2*(3*d*g^5
*e^10 + 5*f*g^4*e^11)*e^(-16))*sqrt(-x^2*e^2 + d^2) - 2/15*(445*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^5*g^5*e^(-2)/
x - 665*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^5*g^5*e^(-4)/x^2 + 405*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^5*g^5*e^(
-6)/x^3 - 90*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^5*g^5*e^(-8)/x^4 - 107*d^5*g^5 - 285*d^4*f*g^4*e + 1200*(d*e +
sqrt(-x^2*e^2 + d^2)*e)*d^4*f*g^4*e^(-1)/x - 1800*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^4*f*g^4*e^(-3)/x^2 + 105
0*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^4*f*g^4*e^(-5)/x^3 - 225*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^4*f*g^4*e^(-7
)/x^4 - 220*d^3*f^2*g^3*e^2 - 1450*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^3*f^2*g^3*e^(-2)/x^2 + 750*(d*e + sqrt(-
x^2*e^2 + d^2)*e)^3*d^3*f^2*g^3*e^(-4)/x^3 - 150*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^3*f^2*g^3*e^(-6)/x^4 + 950
*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^3*f^2*g^3/x - 20*d^2*f^3*g^2*e^3 + 100*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*f^
3*g^2*e/x - 200*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*f^3*g^2*e^(-1)/x^2 + 15*d*f^4*g*e^4 - 75*(d*e + sqrt(-x^2
*e^2 + d^2)*e)*d*f^4*g*e^2/x - 75*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d*f^4*g*e^(-2)/x^3 + 75*(d*e + sqrt(-x^2*e^
2 + d^2)*e)^2*d*f^4*g/x^2 - 7*f^5*e^5 + 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*f^5*e^3/x - 40*(d*e + sqrt(-x^2*e^2
+ d^2)*e)^2*f^5*e/x^2 + 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*f^5*e^(-1)/x^3 - 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)
^4*f^5*e^(-3)/x^4)*e^(-6)/(d^3*((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 1)^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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