3.6.80 \(\int \frac {(d+e x)^3 (f+g x)^4}{(d^2-e^2 x^2)^{7/2}} \, dx\) [580]

Optimal. Leaf size=215 \[ \frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {g^3 (4 e f+3 d g) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5} \]

[Out]

1/5*(d*g+e*f)^4*(e*x+d)^3/d/e^5/(-e^2*x^2+d^2)^(5/2)+2/15*(-9*d*g+e*f)*(d*g+e*f)^3*(e*x+d)^2/d^2/e^5/(-e^2*x^2
+d^2)^(3/2)-g^3*(3*d*g+4*e*f)*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^5+2/15*(d*g+e*f)^2*(36*d^2*g^2-8*d*e*f*g+e^2*
f^2)*(e*x+d)/d^3/e^5/(-e^2*x^2+d^2)^(1/2)+g^4*(-e^2*x^2+d^2)^(1/2)/e^5

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Rubi [A]
time = 0.41, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1649, 655, 223, 209} \begin {gather*} -\frac {g^3 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) (3 d g+4 e f)}{e^5}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}+\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(f + g*x)^4)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((e*f + d*g)^4*(d + e*x)^3)/(5*d*e^5*(d^2 - e^2*x^2)^(5/2)) + (2*(e*f - 9*d*g)*(e*f + d*g)^3*(d + e*x)^2)/(15*
d^2*e^5*(d^2 - e^2*x^2)^(3/2)) + (2*(e*f + d*g)^2*(e^2*f^2 - 8*d*e*f*g + 36*d^2*g^2)*(d + e*x))/(15*d^3*e^5*Sq
rt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/e^5 - (g^3*(4*e*f + 3*d*g)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e
^5

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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-\frac {2 e^4 f^4-12 d e^3 f^3 g-18 d^2 e^2 f^2 g^2-12 d^3 e f g^3-3 d^4 g^4}{e^4}+\frac {5 d g^2 \left (6 e^2 f^2+4 d e f g+d^2 g^2\right ) x}{e^3}+\frac {5 d g^3 (4 e f+d g) x^2}{e^2}+\frac {5 d g^4 x^3}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d+e x) \left (\frac {2 e^4 f^4-12 d e^3 f^3 g+42 d^2 e^2 f^2 g^2+68 d^3 e f g^3+27 d^4 g^4}{e^4}+\frac {30 d^2 g^3 (2 e f+d g) x}{e^3}+\frac {15 d^2 g^4 x^2}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {\frac {15 d^3 g^3 (4 e f+3 d g)}{e^4}+\frac {15 d^3 g^4 x}{e^3}}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {\left (g^3 (4 e f+3 d g)\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^4}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {\left (g^3 (4 e f+3 d g)\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}\\ &=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {g^3 (4 e f+3 d g) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}\\ \end {align*}

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Mathematica [A]
time = 1.29, size = 240, normalized size = 1.12 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (72 d^6 g^4+2 e^6 f^4 x^2+d^5 e g^3 (88 f-171 g x)-6 d e^5 f^3 x (f+2 g x)+3 d^4 e^2 g^2 \left (4 f^2-68 f g x+39 g^2 x^2\right )+d^2 e^4 f^2 \left (7 f^2+36 f g x+42 g^2 x^2\right )-d^3 e^3 g \left (12 f^3+36 f^2 g x-128 f g^2 x^2+15 g^3 x^3\right )\right )}{15 d^3 e^5 (d-e x)^3}+\frac {g^3 (4 e f+3 d g) \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^4 \sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(f + g*x)^4)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(72*d^6*g^4 + 2*e^6*f^4*x^2 + d^5*e*g^3*(88*f - 171*g*x) - 6*d*e^5*f^3*x*(f + 2*g*x) + 3*
d^4*e^2*g^2*(4*f^2 - 68*f*g*x + 39*g^2*x^2) + d^2*e^4*f^2*(7*f^2 + 36*f*g*x + 42*g^2*x^2) - d^3*e^3*g*(12*f^3
+ 36*f^2*g*x - 128*f*g^2*x^2 + 15*g^3*x^3)))/(15*d^3*e^5*(d - e*x)^3) + (g^3*(4*e*f + 3*d*g)*Log[-(Sqrt[-e^2]*
x) + Sqrt[d^2 - e^2*x^2]])/(e^4*Sqrt[-e^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(841\) vs. \(2(199)=398\).
time = 0.11, size = 842, normalized size = 3.92 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(g*x+f)^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

e^3*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*e^2*g^4+4*e^3*f*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^2*e*g^4+12*d*e^2*f*g^3+6*e^3*f^2*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)))+(d^3*g^4+12*d^2*e*f*g^3+18*d*e^2*f^2*g^
2+4*e^3*f^3*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)))))+(4*d^3*f
*g^3+18*d^2*e*f^2*g^2+12*d*e^2*f^3*g+e^3*f^4)*(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))+(6*d^3*f^2*g^2+12*d^2*e*f^3*g+3*d*e^2*f^4)*(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*(4*d^3*f^3*g+3*d^2
*e*f^4)/e^2/(-e^2*x^2+d^2)^(5/2)+d^3*f^4*(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. 1107 vs. \(2 (199) = 398\).
time = 0.55, size = 1107, normalized size = 5.15 \begin {gather*} -\frac {g^{4} x^{6} e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {6 \, d^{2} g^{4} x^{4} e^{\left (-1\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {8 \, d^{4} g^{4} x^{2} e^{\left (-3\right )}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {16 \, d^{6} g^{4} e^{\left (-5\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d^{3} f^{3} g e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f^{4} e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {1}{3} \, {\left (3 \, d g^{4} e^{2} + 4 \, f g^{3} e^{3}\right )} {\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 )} x e^{\left (-2\right )} + \frac {d f^{4} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, {\left (d^{2} g^{4} e + 4 \, d f g^{3} e^{2} + 2 \, f^{2} 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^{2} g^{4} e + 4 \, d f g^{3} e^{2} + 2 \, f^{2} 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^{2} g^{4} e + 4 \, d f g^{3} e^{2} + 2 \, f^{2} g^{2} e^{3}\right )} d^{4} e^{\left (-6\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, {\left (3 \, d g^{4} e^{2} + 4 \, f g^{3} e^{3}\right )} d^{2} x e^{\left (-6\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {1}{15} \, {\left (3 \, d g^{4} e^{2} + 4 \, f g^{3} e^{3}\right )} {\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 )} x - {\left (3 \, d g^{4} e^{2} + 4 \, f g^{3} e^{3}\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-7\right )} + \frac {4 \, f^{4} x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {{\left (d^{3} g^{4} + 12 \, d^{2} f g^{3} e + 18 \, d f^{2} g^{2} e^{2} + 4 \, f^{3} g e^{3}\right )} x^{3} e^{\left (-2\right )}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {3 \, {\left (d^{3} g^{4} + 12 \, d^{2} f g^{3} e + 18 \, d f^{2} g^{2} e^{2} + 4 \, f^{3} g e^{3}\right )} d^{2} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {7 \, {\left (3 \, d g^{4} e^{2} + 4 \, f g^{3} e^{3}\right )} x e^{\left (-6\right )}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}}} + \frac {8 \, f^{4} x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}} + \frac {{\left (4 \, d^{3} f g^{3} + 18 \, d^{2} f^{2} g^{2} e + 12 \, d f^{3} g e^{2} + f^{4} e^{3}\right )} x^{2} e^{\left (-2\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {2 \, {\left (4 \, d^{3} f g^{3} + 18 \, d^{2} f^{2} g^{2} e + 12 \, d f^{3} g e^{2} + f^{4} e^{3}\right )} d^{2} e^{\left (-4\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (d^{3} g^{4} + 12 \, d^{2} f g^{3} e + 18 \, d f^{2} g^{2} e^{2} + 4 \, f^{3} g e^{3}\right )} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (2 \, d^{3} f^{2} g^{2} + 4 \, d^{2} f^{3} g e + d f^{4} e^{2}\right )} x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (d^{3} g^{4} + 12 \, d^{2} f g^{3} e + 18 \, d f^{2} g^{2} e^{2} + 4 \, f^{3} g e^{3}\right )} x e^{\left (-4\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} - \frac {{\left (2 \, d^{3} f^{2} g^{2} + 4 \, d^{2} f^{3} g e + d f^{4} 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 (2 \, d^{3} f^{2} g^{2} + 4 \, d^{2} f^{3} g e + d f^{4} 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)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

-g^4*x^6*e/(-x^2*e^2 + d^2)^(5/2) + 6*d^2*g^4*x^4*e^(-1)/(-x^2*e^2 + d^2)^(5/2) - 8*d^4*g^4*x^2*e^(-3)/(-x^2*e
^2 + d^2)^(5/2) + 16/5*d^6*g^4*e^(-5)/(-x^2*e^2 + d^2)^(5/2) + 4/5*d^3*f^3*g*e^(-2)/(-x^2*e^2 + d^2)^(5/2) + 3
/5*d^2*f^4*e^(-1)/(-x^2*e^2 + d^2)^(5/2) - 1/3*(3*d*g^4*e^2 + 4*f*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) + 1/5*d*f^4*x/(-x^2*e^2 + d^2)^(5/2) + 3*(d^2*g^4*e + 4*d*f*
g^3*e^2 + 2*f^2*g^2*e^3)*x^4*e^(-2)/(-x^2*e^2 + d^2)^(5/2) - 4*(d^2*g^4*e + 4*d*f*g^3*e^2 + 2*f^2*g^2*e^3)*d^2
*x^2*e^(-4)/(-x^2*e^2 + d^2)^(5/2) + 8/5*(d^2*g^4*e + 4*d*f*g^3*e^2 + 2*f^2*g^2*e^3)*d^4*e^(-6)/(-x^2*e^2 + d^
2)^(5/2) + 4/15*(3*d*g^4*e^2 + 4*f*g^3*e^3)*d^2*x*e^(-6)/(-x^2*e^2 + d^2)^(3/2) + 1/15*(3*d*g^4*e^2 + 4*f*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*g^4*e^2 + 4*f*g^3*e^3)*arcsin(x*e/d)*e^(-7) + 4/15*f^4*x/((-x^2*e^2 + d^2)^(3/2)*d) +
1/2*(d^3*g^4 + 12*d^2*f*g^3*e + 18*d*f^2*g^2*e^2 + 4*f^3*g*e^3)*x^3*e^(-2)/(-x^2*e^2 + d^2)^(5/2) - 3/10*(d^3*
g^4 + 12*d^2*f*g^3*e + 18*d*f^2*g^2*e^2 + 4*f^3*g*e^3)*d^2*x*e^(-4)/(-x^2*e^2 + d^2)^(5/2) - 7/15*(3*d*g^4*e^2
+ 4*f*g^3*e^3)*x*e^(-6)/sqrt(-x^2*e^2 + d^2) + 8/15*f^4*x/(sqrt(-x^2*e^2 + d^2)*d^3) + 1/3*(4*d^3*f*g^3 + 18*
d^2*f^2*g^2*e + 12*d*f^3*g*e^2 + f^4*e^3)*x^2*e^(-2)/(-x^2*e^2 + d^2)^(5/2) - 2/15*(4*d^3*f*g^3 + 18*d^2*f^2*g
^2*e + 12*d*f^3*g*e^2 + f^4*e^3)*d^2*e^(-4)/(-x^2*e^2 + d^2)^(5/2) + 1/10*(d^3*g^4 + 12*d^2*f*g^3*e + 18*d*f^2
*g^2*e^2 + 4*f^3*g*e^3)*x*e^(-4)/(-x^2*e^2 + d^2)^(3/2) + 3/5*(2*d^3*f^2*g^2 + 4*d^2*f^3*g*e + d*f^4*e^2)*x*e^
(-2)/(-x^2*e^2 + d^2)^(5/2) + 1/5*(d^3*g^4 + 12*d^2*f*g^3*e + 18*d*f^2*g^2*e^2 + 4*f^3*g*e^3)*x*e^(-4)/(sqrt(-
x^2*e^2 + d^2)*d^2) - 1/5*(2*d^3*f^2*g^2 + 4*d^2*f^3*g*e + d*f^4*e^2)*x*e^(-2)/((-x^2*e^2 + d^2)^(3/2)*d^2) -
2/5*(2*d^3*f^2*g^2 + 4*d^2*f^3*g*e + d*f^4*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. 603 vs. \(2 (199) = 398\).
time = 2.65, size = 603, normalized size = 2.80 \begin {gather*} -\frac {72 \, d^{7} g^{4} - 7 \, f^{4} x^{3} e^{7} + 30 \, {\left (3 \, d^{7} g^{4} - 4 \, d^{3} f g^{3} x^{3} e^{4} - 3 \, {\left (d^{4} g^{4} x^{3} - 4 \, d^{4} f g^{3} x^{2}\right )} e^{3} + 3 \, {\left (3 \, d^{5} g^{4} x^{2} - 4 \, d^{5} f g^{3} x\right )} e^{2} - {\left (9 \, d^{6} g^{4} x - 4 \, d^{6} f g^{3}\right )} e\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + 3 \, {\left (4 \, d f^{3} g x^{3} + 7 \, d f^{4} x^{2}\right )} e^{6} - 3 \, {\left (4 \, d^{2} f^{2} g^{2} x^{3} + 12 \, d^{2} f^{3} g x^{2} + 7 \, d^{2} f^{4} x\right )} e^{5} - {\left (88 \, d^{3} f g^{3} x^{3} - 36 \, d^{3} f^{2} g^{2} x^{2} - 36 \, d^{3} f^{3} g x - 7 \, d^{3} f^{4}\right )} e^{4} - 12 \, {\left (6 \, d^{4} g^{4} x^{3} - 22 \, d^{4} f g^{3} x^{2} + 3 \, d^{4} f^{2} g^{2} x + d^{4} f^{3} g\right )} e^{3} + 12 \, {\left (18 \, d^{5} g^{4} x^{2} - 22 \, d^{5} f g^{3} x + d^{5} f^{2} g^{2}\right )} e^{2} - 8 \, {\left (27 \, d^{6} g^{4} x - 11 \, d^{6} f g^{3}\right )} e + {\left (72 \, d^{6} g^{4} + 2 \, f^{4} x^{2} e^{6} - 6 \, {\left (2 \, d f^{3} g x^{2} + d f^{4} x\right )} e^{5} + {\left (42 \, d^{2} f^{2} g^{2} x^{2} + 36 \, d^{2} f^{3} g x + 7 \, d^{2} f^{4}\right )} e^{4} - {\left (15 \, d^{3} g^{4} x^{3} - 128 \, d^{3} f g^{3} x^{2} + 36 \, d^{3} f^{2} g^{2} x + 12 \, d^{3} f^{3} g\right )} e^{3} + 3 \, {\left (39 \, d^{4} g^{4} x^{2} - 68 \, d^{4} f g^{3} x + 4 \, d^{4} f^{2} g^{2}\right )} e^{2} - {\left (171 \, d^{5} g^{4} x - 88 \, d^{5} f g^{3}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{3} e^{8} - 3 \, d^{4} x^{2} e^{7} + 3 \, d^{5} x e^{6} - d^{6} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

-1/15*(72*d^7*g^4 - 7*f^4*x^3*e^7 + 30*(3*d^7*g^4 - 4*d^3*f*g^3*x^3*e^4 - 3*(d^4*g^4*x^3 - 4*d^4*f*g^3*x^2)*e^
3 + 3*(3*d^5*g^4*x^2 - 4*d^5*f*g^3*x)*e^2 - (9*d^6*g^4*x - 4*d^6*f*g^3)*e)*arctan(-(d - sqrt(-x^2*e^2 + d^2))*
e^(-1)/x) + 3*(4*d*f^3*g*x^3 + 7*d*f^4*x^2)*e^6 - 3*(4*d^2*f^2*g^2*x^3 + 12*d^2*f^3*g*x^2 + 7*d^2*f^4*x)*e^5 -
(88*d^3*f*g^3*x^3 - 36*d^3*f^2*g^2*x^2 - 36*d^3*f^3*g*x - 7*d^3*f^4)*e^4 - 12*(6*d^4*g^4*x^3 - 22*d^4*f*g^3*x
^2 + 3*d^4*f^2*g^2*x + d^4*f^3*g)*e^3 + 12*(18*d^5*g^4*x^2 - 22*d^5*f*g^3*x + d^5*f^2*g^2)*e^2 - 8*(27*d^6*g^4
*x - 11*d^6*f*g^3)*e + (72*d^6*g^4 + 2*f^4*x^2*e^6 - 6*(2*d*f^3*g*x^2 + d*f^4*x)*e^5 + (42*d^2*f^2*g^2*x^2 + 3
6*d^2*f^3*g*x + 7*d^2*f^4)*e^4 - (15*d^3*g^4*x^3 - 128*d^3*f*g^3*x^2 + 36*d^3*f^2*g^2*x + 12*d^3*f^3*g)*e^3 +
3*(39*d^4*g^4*x^2 - 68*d^4*f*g^3*x + 4*d^4*f^2*g^2)*e^2 - (171*d^5*g^4*x - 88*d^5*f*g^3)*e)*sqrt(-x^2*e^2 + d^
2))/(d^3*x^3*e^8 - 3*d^4*x^2*e^7 + 3*d^5*x*e^6 - d^6*e^5)

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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 )^{4}}{\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)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**3*(f + g*x)**4/(-(-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. 729 vs. \(2 (199) = 398\).
time = 2.17, size = 729, normalized size = 3.39 \begin {gather*} \sqrt {-x^{2} e^{2} + d^{2}} g^{4} e^{\left (-5\right )} - {\left (3 \, d g^{4} + 4 \, f g^{3} e\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) - \frac {2 \, {\left (\frac {240 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} g^{4} e^{\left (-2\right )}}{x} - \frac {360 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} g^{4} e^{\left (-4\right )}}{x^{2}} + \frac {210 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} g^{4} e^{\left (-6\right )}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} g^{4} e^{\left (-8\right )}}{x^{4}} - 57 \, d^{4} g^{4} - 88 \, d^{3} f g^{3} e + \frac {380 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} f g^{3} e^{\left (-1\right )}}{x} - \frac {580 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} f g^{3} e^{\left (-3\right )}}{x^{2}} + \frac {300 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} f g^{3} e^{\left (-5\right )}}{x^{3}} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} f g^{3} e^{\left (-7\right )}}{x^{4}} - 12 \, d^{2} f^{2} g^{2} e^{2} - \frac {120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} f^{2} g^{2} e^{\left (-2\right )}}{x^{2}} + \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} f^{2} g^{2}}{x} + 12 \, d f^{3} g e^{3} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f^{3} g e}{x} + \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f^{3} g e^{\left (-1\right )}}{x^{2}} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f^{3} g e^{\left (-3\right )}}{x^{3}} - 7 \, f^{4} e^{4} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{4} e^{2}}{x} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{4} e^{\left (-2\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{4} e^{\left (-4\right )}}{x^{4}} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{4}}{x^{2}}\right )} e^{\left (-5\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)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

sqrt(-x^2*e^2 + d^2)*g^4*e^(-5) - (3*d*g^4 + 4*f*g^3*e)*arcsin(x*e/d)*e^(-5)*sgn(d) - 2/15*(240*(d*e + sqrt(-x
^2*e^2 + d^2)*e)*d^4*g^4*e^(-2)/x - 360*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^4*g^4*e^(-4)/x^2 + 210*(d*e + sqrt(
-x^2*e^2 + d^2)*e)^3*d^4*g^4*e^(-6)/x^3 - 45*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^4*g^4*e^(-8)/x^4 - 57*d^4*g^4
- 88*d^3*f*g^3*e + 380*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^3*f*g^3*e^(-1)/x - 580*(d*e + sqrt(-x^2*e^2 + d^2)*e)^
2*d^3*f*g^3*e^(-3)/x^2 + 300*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^3*f*g^3*e^(-5)/x^3 - 60*(d*e + sqrt(-x^2*e^2 +
d^2)*e)^4*d^3*f*g^3*e^(-7)/x^4 - 12*d^2*f^2*g^2*e^2 - 120*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*f^2*g^2*e^(-2)
/x^2 + 60*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*f^2*g^2/x + 12*d*f^3*g*e^3 - 60*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d*
f^3*g*e/x + 60*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d*f^3*g*e^(-1)/x^2 - 60*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d*f^3
*g*e^(-3)/x^3 - 7*f^4*e^4 + 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*f^4*e^2/x + 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*
f^4*e^(-2)/x^3 - 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*f^4*e^(-4)/x^4 - 40*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*f^4/
x^2)*e^(-5)/(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 )}^4\,{\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)^4*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x)

[Out]

int(((f + g*x)^4*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2), x)

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