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

Optimal result

Integrand size = 31, antiderivative size = 215 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {g^3 (4 e f+3 d g) \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5} \]

[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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1649, 655, 223, 209} \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {g^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) (3 d g+4 e f)}{e^5}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}+\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (72 d^6 g^4+2 e^6 f^4 x^2+d^5 e g^3 (88 f-171 g x)-6 d e^5 f^3 x (f+2 g x)+3 d^4 e^2 g^2 \left (4 f^2-68 f g x+39 g^2 x^2\right )+d^2 e^4 f^2 \left (7 f^2+36 f g x+42 g^2 x^2\right )-d^3 e^3 g \left (12 f^3+36 f^2 g x-128 f g^2 x^2+15 g^3 x^3\right )\right )}{15 d^3 e^5 (d-e x)^3}+\frac {g^3 (4 e f+3 d g) \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^4 \sqrt {-e^2}} \]

[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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(480\) vs. \(2(199)=398\).

Time = 0.90 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.24

method	result	size
risch	\(\frac {g^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{e^{5}}-\frac {\frac {\left (3 d g +4 e f \right ) g^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e \sqrt {e^{2}}}+\frac {6 g^{2} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{e^{3} d \left (x -\frac {d}{e}\right )}+\frac {4 g \left (d^{3} g^{3}+3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{e^{3}}+\frac {\left (g^{4} d^{4}+4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}+4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{5 d e \left (x -\frac {d}{e}\right )^{3}}-\frac {2 e \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{5 d}\right )}{e^{4}}}{e^{3}}\)	\(481\)
default	\(d^{3} f^{4} \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^{4} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \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 )}{e^{2}}\right )+\left (3 d \,e^{2} g^{4}+4 e^{3} f \,g^{3}\right ) \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 )+\frac {4 d^{3} f^{3} g +3 d^{2} e \,f^{4}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\left (3 d^{2} e \,g^{4}+12 d \,e^{2} f \,g^{3}+6 e^{3} f^{2} 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 (6 d^{3} f^{2} g^{2}+12 d^{2} e \,f^{3} g +3 d \,e^{2} f^{4}\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^{4}+12 d^{2} e f \,g^{3}+18 d \,e^{2} f^{2} g^{2}+4 e^{3} f^{3} 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 (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}\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 )\)	\(842\)

[In]

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

[Out]

g^4*(-e^2*x^2+d^2)^(1/2)/e^5-1/e^3*((3*d*g+4*e*f)*g^3/e/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
+6*g^2/e^3*(d^2*g^2+2*d*e*f*g+e^2*f^2)/d/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)+4/e^3*g*(d^3*g^3+3*d^2*e
*f*g^2+3*d*e^2*f^2*g+e^3*f^3)*(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))+1/e^4*(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)*(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. 624 vs. \(2 (199) = 398\).

Time = 0.40 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.90 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 \, d^{3} e^{4} f^{4} - 12 \, d^{4} e^{3} f^{3} g + 12 \, d^{5} e^{2} f^{2} g^{2} + 88 \, d^{6} e f g^{3} + 72 \, d^{7} g^{4} - {\left (7 \, e^{7} f^{4} - 12 \, d e^{6} f^{3} g + 12 \, d^{2} e^{5} f^{2} g^{2} + 88 \, d^{3} e^{4} f g^{3} + 72 \, d^{4} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (7 \, d e^{6} f^{4} - 12 \, d^{2} e^{5} f^{3} g + 12 \, d^{3} e^{4} f^{2} g^{2} + 88 \, d^{4} e^{3} f g^{3} + 72 \, d^{5} e^{2} g^{4}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{5} f^{4} - 12 \, d^{3} e^{4} f^{3} g + 12 \, d^{4} e^{3} f^{2} g^{2} + 88 \, d^{5} e^{2} f g^{3} + 72 \, d^{6} e g^{4}\right )} x + 30 \, {\left (4 \, d^{6} e f g^{3} + 3 \, d^{7} g^{4} - {\left (4 \, d^{3} e^{4} f g^{3} + 3 \, d^{4} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (4 \, d^{4} e^{3} f g^{3} + 3 \, d^{5} e^{2} g^{4}\right )} x^{2} - 3 \, {\left (4 \, d^{5} e^{2} f g^{3} + 3 \, d^{6} e g^{4}\right )} x\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, d^{3} e^{3} g^{4} x^{3} - 7 \, d^{2} e^{4} f^{4} + 12 \, d^{3} e^{3} f^{3} g - 12 \, d^{4} e^{2} f^{2} g^{2} - 88 \, d^{5} e f g^{3} - 72 \, d^{6} g^{4} - {\left (2 \, e^{6} f^{4} - 12 \, d e^{5} f^{3} g + 42 \, d^{2} e^{4} f^{2} g^{2} + 128 \, d^{3} e^{3} f g^{3} + 117 \, d^{4} e^{2} g^{4}\right )} x^{2} + 3 \, {\left (2 \, d e^{5} f^{4} - 12 \, d^{2} e^{4} f^{3} g + 12 \, d^{3} e^{3} f^{2} g^{2} + 68 \, d^{4} e^{2} f g^{3} + 57 \, d^{5} e g^{4}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{8} x^{3} - 3 \, d^{4} e^{7} x^{2} + 3 \, d^{5} e^{6} x - d^{6} e^{5}\right )}} \]

[In]

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

[Out]

-1/15*(7*d^3*e^4*f^4 - 12*d^4*e^3*f^3*g + 12*d^5*e^2*f^2*g^2 + 88*d^6*e*f*g^3 + 72*d^7*g^4 - (7*e^7*f^4 - 12*d
*e^6*f^3*g + 12*d^2*e^5*f^2*g^2 + 88*d^3*e^4*f*g^3 + 72*d^4*e^3*g^4)*x^3 + 3*(7*d*e^6*f^4 - 12*d^2*e^5*f^3*g +
12*d^3*e^4*f^2*g^2 + 88*d^4*e^3*f*g^3 + 72*d^5*e^2*g^4)*x^2 - 3*(7*d^2*e^5*f^4 - 12*d^3*e^4*f^3*g + 12*d^4*e^
3*f^2*g^2 + 88*d^5*e^2*f*g^3 + 72*d^6*e*g^4)*x + 30*(4*d^6*e*f*g^3 + 3*d^7*g^4 - (4*d^3*e^4*f*g^3 + 3*d^4*e^3*
g^4)*x^3 + 3*(4*d^4*e^3*f*g^3 + 3*d^5*e^2*g^4)*x^2 - 3*(4*d^5*e^2*f*g^3 + 3*d^6*e*g^4)*x)*arctan(-(d - sqrt(-e
^2*x^2 + d^2))/(e*x)) - (15*d^3*e^3*g^4*x^3 - 7*d^2*e^4*f^4 + 12*d^3*e^3*f^3*g - 12*d^4*e^2*f^2*g^2 - 88*d^5*e
*f*g^3 - 72*d^6*g^4 - (2*e^6*f^4 - 12*d*e^5*f^3*g + 42*d^2*e^4*f^2*g^2 + 128*d^3*e^3*f*g^3 + 117*d^4*e^2*g^4)*
x^2 + 3*(2*d*e^5*f^4 - 12*d^2*e^4*f^3*g + 12*d^3*e^3*f^2*g^2 + 68*d^4*e^2*f*g^3 + 57*d^5*e*g^4)*x)*sqrt(-e^2*x
^2 + d^2))/(d^3*e^8*x^3 - 3*d^4*e^7*x^2 + 3*d^5*e^6*x - d^6*e^5)

Sympy [F]

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

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (199) = 398\).

Time = 0.29 (sec) , antiderivative size = 1190, normalized size of antiderivative = 5.53 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-e*g^4*x^6/(-e^2*x^2 + d^2)^(5/2) + 6*d^2*g^4*x^4/((-e^2*x^2 + d^2)^(5/2)*e) - 8*d^4*g^4*x^2/((-e^2*x^2 + d^2)
^(5/2)*e^3) + 1/5*d*f^4*x/(-e^2*x^2 + d^2)^(5/2) + 1/15*(4*e^3*f*g^3 + 3*d*e^2*g^4)*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)) + 3/5*d^2*f^4/((-
e^2*x^2 + d^2)^(5/2)*e) + 4/5*d^3*f^3*g/((-e^2*x^2 + d^2)^(5/2)*e^2) + 16/5*d^6*g^4/((-e^2*x^2 + d^2)^(5/2)*e^
5) + 4/15*f^4*x/((-e^2*x^2 + d^2)^(3/2)*d) + 8/15*f^4*x/(sqrt(-e^2*x^2 + d^2)*d^3) - 1/3*(4*e^3*f*g^3 + 3*d*e^
2*g^4)*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 + 3*(2*e^3*f^2*g^2 + 4*
d*e^2*f*g^3 + d^2*e*g^4)*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) + 1/2*(4*e^3*f^3*g + 18*d*e^2*f^2*g^2 + 12*d^2*e*f*g
^3 + d^3*g^4)*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 4*(2*e^3*f^2*g^2 + 4*d*e^2*f*g^3 + d^2*e*g^4)*d^2*x^2/((-e^2*
x^2 + d^2)^(5/2)*e^4) + 1/3*(e^3*f^4 + 12*d*e^2*f^3*g + 18*d^2*e*f^2*g^2 + 4*d^3*f*g^3)*x^2/((-e^2*x^2 + d^2)^
(5/2)*e^2) - 3/10*(4*e^3*f^3*g + 18*d*e^2*f^2*g^2 + 12*d^2*e*f*g^3 + d^3*g^4)*d^2*x/((-e^2*x^2 + d^2)^(5/2)*e^
4) + 3/5*(d*e^2*f^4 + 4*d^2*e*f^3*g + 2*d^3*f^2*g^2)*x/((-e^2*x^2 + d^2)^(5/2)*e^2) + 8/5*(2*e^3*f^2*g^2 + 4*d
*e^2*f*g^3 + d^2*e*g^4)*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6) - 2/15*(e^3*f^4 + 12*d*e^2*f^3*g + 18*d^2*e*f^2*g^2 +
4*d^3*f*g^3)*d^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 4/15*(4*e^3*f*g^3 + 3*d*e^2*g^4)*d^2*x/((-e^2*x^2 + d^2)^(3/2
)*e^6) + 1/10*(4*e^3*f^3*g + 18*d*e^2*f^2*g^2 + 12*d^2*e*f*g^3 + d^3*g^4)*x/((-e^2*x^2 + d^2)^(3/2)*e^4) - 1/5
*(d*e^2*f^4 + 4*d^2*e*f^3*g + 2*d^3*f^2*g^2)*x/((-e^2*x^2 + d^2)^(3/2)*d^2*e^2) - 7/15*(4*e^3*f*g^3 + 3*d*e^2*
g^4)*x/(sqrt(-e^2*x^2 + d^2)*e^6) + 1/5*(4*e^3*f^3*g + 18*d*e^2*f^2*g^2 + 12*d^2*e*f*g^3 + d^3*g^4)*x/(sqrt(-e
^2*x^2 + d^2)*d^2*e^4) - 2/5*(d*e^2*f^4 + 4*d^2*e*f^3*g + 2*d^3*f^2*g^2)*x/(sqrt(-e^2*x^2 + d^2)*d^4*e^2) - (4
*e^3*f*g^3 + 3*d*e^2*g^4)*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. 757 vs. \(2 (199) = 398\).

Time = 0.31 (sec) , antiderivative size = 757, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {-e^{2} x^{2} + d^{2}} g^{4}}{e^{5}} - \frac {{\left (4 \, e f g^{3} + 3 \, d g^{4}\right )} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{e^{4} {\left | e \right |}} + \frac {2 \, {\left (7 \, e^{4} f^{4} - 12 \, d e^{3} f^{3} g + 12 \, d^{2} e^{2} f^{2} g^{2} + 88 \, d^{3} e f g^{3} + 57 \, d^{4} g^{4} - \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{2} f^{4}}{x} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e f^{3} g}{x} - \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} f^{2} g^{2}}{x} - \frac {380 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3} f g^{3}}{e x} - \frac {240 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} g^{4}}{e^{2} x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} f^{4}}{x^{2}} - \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d f^{3} g}{e x^{2}} + \frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} f^{2} g^{2}}{e^{2} x^{2}} + \frac {580 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{3} f g^{3}}{e^{3} x^{2}} + \frac {360 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} g^{4}}{e^{4} x^{2}} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} f^{4}}{e^{2} x^{3}} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d f^{3} g}{e^{3} x^{3}} - \frac {300 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3} f g^{3}}{e^{5} x^{3}} - \frac {210 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4} g^{4}}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} f^{4}}{e^{4} x^{4}} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3} f g^{3}}{e^{7} x^{4}} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{4} g^{4}}{e^{8} x^{4}}\right )}}{15 \, d^{3} e^{4} {\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)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^4\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

[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)