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

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

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Rubi [A]  time = 0.97, 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 = {1635, 1815, 641, 217, 203} \begin {gather*} \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}}-\frac {g^3 \left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^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} \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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

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 641

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 1635

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 1815

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 ) \operatorname {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 = 0.97, size = 193, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (\frac {2 (2 e f-23 d g) (d g+e f)^4}{d^2 (d-e x)^2}+\frac {2 (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d^3 (d-e x)}+30 g^4 (3 d g+5 e f)+\frac {6 (d g+e f)^5}{d (d-e x)^3}+15 e g^5 x\right )-15 g^3 \left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{30 e^6} \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]

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

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IntegrateAlgebraic [A]  time = 1.98, size = 385, normalized size = 1.43 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (304 d^7 g^5+720 d^6 e f g^4-717 d^6 e g^5 x+440 d^5 e^2 f^2 g^3-1710 d^5 e^2 f g^4 x+479 d^5 e^2 g^5 x^2+40 d^4 e^3 f^3 g^2-1020 d^4 e^3 f^2 g^3 x+1170 d^4 e^3 f g^4 x^2-45 d^4 e^3 g^5 x^3-30 d^3 e^4 f^4 g-120 d^3 e^4 f^3 g^2 x+640 d^3 e^4 f^2 g^3 x^2-150 d^3 e^4 f g^4 x^3-15 d^3 e^4 g^5 x^4+14 d^2 e^5 f^5+90 d^2 e^5 f^4 g x+140 d^2 e^5 f^3 g^2 x^2-12 d e^6 f^5 x-30 d e^6 f^4 g x^2+4 e^7 f^5 x^2\right )}{30 d^3 e^6 (d-e x)^3}-\frac {\sqrt {-e^2} \left (13 d^2 g^5+30 d e f g^4+20 e^2 f^2 g^3\right ) \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{2 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((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]*(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 - 12*d*e^6*f^5*x + 90*d^2*e^5*f^4*g*x - 120*d^3*e^4*f^3*g^2*x - 1020*d^4*e^3*f^2*g^3*x -
1710*d^5*e^2*f*g^4*x - 717*d^6*e*g^5*x + 4*e^7*f^5*x^2 - 30*d*e^6*f^4*g*x^2 + 140*d^2*e^5*f^3*g^2*x^2 + 640*d^
3*e^4*f^2*g^3*x^2 + 1170*d^4*e^3*f*g^4*x^2 + 479*d^5*e^2*g^5*x^2 - 150*d^3*e^4*f*g^4*x^3 - 45*d^4*e^3*g^5*x^3
- 15*d^3*e^4*g^5*x^4))/(30*d^3*e^6*(d - e*x)^3) - (Sqrt[-e^2]*(20*e^2*f^2*g^3 + 30*d*e*f*g^4 + 13*d^2*g^5)*Log
[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(2*e^7)

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fricas [B]  time = 0.49, size = 807, normalized size = 3.00 \begin {gather*} -\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 )}} \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*(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)

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giac [B]  time = 0.66, size = 537, normalized size = 2.00 \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}\relax (d) + \frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left ({\left ({\left (15 \, {\left (g^{5} x e + \frac {2 \, {\left (3 \, d^{5} g^{5} e^{12} + 5 \, d^{4} f g^{4} e^{13}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {{\left (299 \, d^{6} g^{5} e^{11} + 720 \, d^{5} f g^{4} e^{12} + 640 \, d^{4} f^{2} g^{3} e^{13} + 140 \, d^{3} f^{3} g^{2} e^{14} - 30 \, d^{2} f^{4} g e^{15} + 4 \, d f^{5} e^{16}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {30 \, {\left (19 \, d^{7} g^{5} e^{10} + 45 \, d^{6} f g^{4} e^{11} + 30 \, d^{5} f^{2} g^{3} e^{12} + 10 \, d^{4} f^{3} g^{2} e^{13}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x + \frac {5 \, {\left (91 \, d^{8} g^{5} e^{9} + 210 \, d^{7} f g^{4} e^{10} + 140 \, d^{6} f^{2} g^{3} e^{11} - 20 \, d^{5} f^{3} g^{2} e^{12} - 30 \, d^{4} f^{4} g e^{13} + 2 \, d^{3} f^{5} e^{14}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x + \frac {10 \, {\left (76 \, d^{9} g^{5} e^{8} + 180 \, d^{8} f g^{4} e^{9} + 110 \, d^{7} f^{2} g^{3} e^{10} + 10 \, d^{6} f^{3} g^{2} e^{11} - 15 \, d^{5} f^{4} g e^{12} - d^{4} f^{5} e^{13}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {15 \, {\left (13 \, d^{10} g^{5} e^{7} + 30 \, d^{9} f g^{4} e^{8} + 20 \, d^{8} f^{2} g^{3} e^{9} + 2 \, d^{5} f^{5} e^{12}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {2 \, {\left (152 \, d^{11} g^{5} e^{6} + 360 \, d^{10} f g^{4} e^{7} + 220 \, d^{9} f^{2} g^{3} e^{8} + 20 \, d^{8} f^{3} g^{2} e^{9} - 15 \, d^{7} f^{4} g e^{10} + 7 \, d^{6} f^{5} e^{11}\right )} e^{\left (-12\right )}}{d^{4}}\right )}}{30 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \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) + 1/30*sqrt(-x^2*e^2 + d^2)*((((
((15*(g^5*x*e + 2*(3*d^5*g^5*e^12 + 5*d^4*f*g^4*e^13)*e^(-12)/d^4)*x - (299*d^6*g^5*e^11 + 720*d^5*f*g^4*e^12
+ 640*d^4*f^2*g^3*e^13 + 140*d^3*f^3*g^2*e^14 - 30*d^2*f^4*g*e^15 + 4*d*f^5*e^16)*e^(-12)/d^4)*x - 30*(19*d^7*
g^5*e^10 + 45*d^6*f*g^4*e^11 + 30*d^5*f^2*g^3*e^12 + 10*d^4*f^3*g^2*e^13)*e^(-12)/d^4)*x + 5*(91*d^8*g^5*e^9 +
210*d^7*f*g^4*e^10 + 140*d^6*f^2*g^3*e^11 - 20*d^5*f^3*g^2*e^12 - 30*d^4*f^4*g*e^13 + 2*d^3*f^5*e^14)*e^(-12)
/d^4)*x + 10*(76*d^9*g^5*e^8 + 180*d^8*f*g^4*e^9 + 110*d^7*f^2*g^3*e^10 + 10*d^6*f^3*g^2*e^11 - 15*d^5*f^4*g*e
^12 - d^4*f^5*e^13)*e^(-12)/d^4)*x - 15*(13*d^10*g^5*e^7 + 30*d^9*f*g^4*e^8 + 20*d^8*f^2*g^3*e^9 + 2*d^5*f^5*e
^12)*e^(-12)/d^4)*x - 2*(152*d^11*g^5*e^6 + 360*d^10*f*g^4*e^7 + 220*d^9*f^2*g^3*e^8 + 20*d^8*f^3*g^2*e^9 - 15
*d^7*f^4*g*e^10 + 7*d^6*f^5*e^11)*e^(-12)/d^4)/(x^2*e^2 - d^2)^3

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maple [B]  time = 0.06, size = 1308, normalized size = 4.86

result 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)

[Out]

3/2*d^2/e*x/(-e^2*x^2+d^2)^(5/2)*f^4*g+1/2/e^4*x/(-e^2*x^2+d^2)^(3/2)*d^3*f*g^4+3/e^3*x/(-e^2*x^2+d^2)^(3/2)*d
^2*f^2*g^3+7/3/e^2*x/(-e^2*x^2+d^2)^(3/2)*d*f^3*g^2-5/e^2*x^3/(-e^2*x^2+d^2)^(3/2)*d*f*g^4+16/e^4*x/(-e^2*x^2+
d^2)^(1/2)*d*f*g^4-15/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)*d*f*g^4-110/3*d^3/e^2*x^2/(-e
^2*x^2+d^2)^(5/2)*f^2*g^3-10/3*d^2/e*x^2/(-e^2*x^2+d^2)^(5/2)*f^3*g^2+5/2*x^3/e^2/(-e^2*x^2+d^2)^(5/2)*d^3*f*g
^4+15*x^3/e/(-e^2*x^2+d^2)^(5/2)*d^2*f^2*g^3-3/2*d^5/e^4*x/(-e^2*x^2+d^2)^(5/2)*f*g^4+45*d^2/e*x^4/(-e^2*x^2+d
^2)^(5/2)*f*g^4-60*d^4/e^3*x^2/(-e^2*x^2+d^2)^(5/2)*f*g^4+14/3/d/e^2*x/(-e^2*x^2+d^2)^(1/2)*f^3*g^2-1/d^2/e*x/
(-e^2*x^2+d^2)^(1/2)*f^4*g-9*d^4/e^3*x/(-e^2*x^2+d^2)^(5/2)*f^2*g^3-7*d^3/e^2*x/(-e^2*x^2+d^2)^(5/2)*f^3*g^2-1
/2*e*g^5*x^7/(-e^2*x^2+d^2)^(5/2)+7/15*d^2/e/(-e^2*x^2+d^2)^(5/2)*f^5+4/5*x/(-e^2*x^2+d^2)^(5/2)*d*f^5+1/15/d*
x/(-e^2*x^2+d^2)^(3/2)*f^5+2/15/d^3*x/(-e^2*x^2+d^2)^(1/2)*f^5+1/3*x^2*e/(-e^2*x^2+d^2)^(5/2)*f^5-3*x^6/(-e^2*
x^2+d^2)^(5/2)*d*g^5+152/15*d^7/e^6/(-e^2*x^2+d^2)^(5/2)*g^5+5/2*x^3*e/(-e^2*x^2+d^2)^(5/2)*f^4*g-1/2/e*x/(-e^
2*x^2+d^2)^(3/2)*f^4*g+5*x^2/(-e^2*x^2+d^2)^(5/2)*d*f^4*g-d^3/e^2/(-e^2*x^2+d^2)^(5/2)*f^4*g-5*x^6*e/(-e^2*x^2
+d^2)^(5/2)*f*g^4+19*d^3/e^2*x^4/(-e^2*x^2+d^2)^(5/2)*g^5-76/3*d^5/e^4*x^2/(-e^2*x^2+d^2)^(5/2)*g^5+24*d^6/e^5
/(-e^2*x^2+d^2)^(5/2)*f*g^4+3*x^5/(-e^2*x^2+d^2)^(5/2)*d*f*g^4+2*x^5*e/(-e^2*x^2+d^2)^(5/2)*f^2*g^3-10/3/e*x^3
/(-e^2*x^2+d^2)^(3/2)*f^2*g^3+16/e^3*x/(-e^2*x^2+d^2)^(1/2)*f^2*g^3-10/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^
2*x^2+d^2)^(1/2)*x)*f^2*g^3+30*x^4/(-e^2*x^2+d^2)^(5/2)*d*f^2*g^3+10*x^4*e/(-e^2*x^2+d^2)^(5/2)*f^3*g^2+13/10/
e*g^5*d^2*x^5/(-e^2*x^2+d^2)^(5/2)-13/6/e^3*g^5*d^2*x^3/(-e^2*x^2+d^2)^(3/2)+13/2/e^5*g^5*d^2*x/(-e^2*x^2+d^2)
^(1/2)-13/2/e^5*g^5*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)+44/3*d^5/e^4/(-e^2*x^2+d^2)^(5/
2)*f^2*g^3+4/3*d^4/e^3/(-e^2*x^2+d^2)^(5/2)*f^3*g^2+15*x^3/(-e^2*x^2+d^2)^(5/2)*d*f^3*g^2

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maxima [B]  time = 1.05, size = 1579, normalized size = 5.87

result too large to display

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*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*x/d)/e^6 - 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 + 3
0*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*x/d)/e^7

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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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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