3.6.82 \(\int \frac {(d+e x)^3 (f+g x)^2}{(d^2-e^2 x^2)^{7/2}} \, dx\) [582]

Optimal. Leaf size=145 \[ \frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) (d+e x)}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1649, 803, 651} \begin {gather*} \frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x)^2 (e f-4 d g) (d g+e f)}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x) \left (7 d^2 g^2-6 d e f g+2 e^2 f^2\right )}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*f + d*g)^2*(d + e*x)^3)/(5*d*e^3*(d^2 - e^2*x^2)^(5/2)) + (2*(e*f - 4*d*g)*(e*f + d*g)*(d + e*x)^2)/(15*d^
2*e^3*(d^2 - e^2*x^2)^(3/2)) + ((2*e^2*f^2 - 6*d*e*f*g + 7*d^2*g^2)*(d + e*x))/(15*d^3*e^3*Sqrt[d^2 - e^2*x^2]
)

Rule 651

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[((-a)*e + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rule 803

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g + e*f)*(
d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(p + 1))), x] - Dist[e*((m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))
), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0]
&& LtQ[p, -1] && GtQ[m, 0]

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)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-2 f^2+\frac {6 d f g}{e}+\frac {3 d^2 g^2}{e^2}+\frac {5 d g^2 x}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2 e^2}\\ &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) (d+e x)}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 105, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (2 d^4 g^2+2 e^4 f^2 x^2-6 d^3 e g (f+g x)-6 d e^3 f x (f+g x)+d^2 e^2 \left (7 f^2+18 f g x+7 g^2 x^2\right )\right )}{15 d^3 e^3 (d-e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(2*d^4*g^2 + 2*e^4*f^2*x^2 - 6*d^3*e*g*(f + g*x) - 6*d*e^3*f*x*(f + g*x) + d^2*e^2*(7*f^2
+ 18*f*g*x + 7*g^2*x^2)))/(15*d^3*e^3*(d - e*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(531\) vs. \(2(133)=266\).
time = 0.09, size = 532, normalized size = 3.67

method	result	size
trager	\(\frac {\left (7 d^{2} e^{2} g^{2} x^{2}-6 d \,e^{3} f g \,x^{2}+2 e^{4} f^{2} x^{2}-6 d^{3} e \,g^{2} x +18 d^{2} e^{2} f g x -6 d \,e^{3} f^{2} x +2 d^{4} g^{2}-6 f g \,d^{3} e +7 d^{2} e^{2} f^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} e^{3} \left (-e x +d \right )^{3}}\)	\(126\)
gosper	\(\frac {\left (e x +d \right )^{4} \left (-e x +d \right ) \left (7 d^{2} e^{2} g^{2} x^{2}-6 d \,e^{3} f g \,x^{2}+2 e^{4} f^{2} x^{2}-6 d^{3} e \,g^{2} x +18 d^{2} e^{2} f g x -6 d \,e^{3} f^{2} x +2 d^{4} g^{2}-6 f g \,d^{3} e +7 d^{2} e^{2} f^{2}\right )}{15 d^{3} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(131\)
default	\(g^{2} e^{3} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+\left (3 e^{2} d \,g^{2}+2 e^{3} f g \right ) \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+\left (3 g^{2} d^{2} e +6 f g d \,e^{2}+f^{2} e^{3}\right ) \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+\left (d^{3} g^{2}+6 d^{2} e f g +3 d \,e^{2} f^{2}\right ) \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {2 d^{3} f g +3 e \,d^{2} f^{2}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{3} f^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\)	\(532\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g^2*e^3*(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^2+2*e^3*f*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)))))+(3*d^2*e*g^2+6*d*e^2*f*g+e^3*f^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^2+6*d^2*e*f*g+3*d*e^2*f^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))))+1/5*(2*d^3*f*g+3*d^2*e*f^2)/e^2/(-
e^2*x^2+d^2)^(5/2)+d^3*f^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)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (134) = 268\).
time = 0.32, size = 544, normalized size = 3.75 \begin {gather*} \frac {g^{2} x^{4} e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {4 \, d^{2} g^{2} x^{2} e^{\left (-1\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, d^{4} g^{2} e^{\left (-3\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d^{3} f g e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f^{2} e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d g^{2} e^{2} + 2 \, f 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 (3 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} d^{2} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d f^{2} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d^{2} g^{2} e + 6 \, d f g e^{2} + f^{2} e^{3}\right )} x^{2} e^{\left (-2\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {2 \, {\left (3 \, d^{2} g^{2} e + 6 \, d f g e^{2} + f^{2} e^{3}\right )} d^{2} e^{\left (-4\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {4 \, f^{2} x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {{\left (d^{3} g^{2} + 6 \, d^{2} f g e + 3 \, d f^{2} e^{2}\right )} x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} x e^{\left (-4\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} + \frac {8 \, f^{2} x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}} - \frac {{\left (d^{3} g^{2} + 6 \, d^{2} f g e + 3 \, d f^{2} e^{2}\right )} x e^{\left (-2\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} - \frac {2 \, {\left (d^{3} g^{2} + 6 \, d^{2} f g e + 3 \, d f^{2} e^{2}\right )} x e^{\left (-2\right )}}{15 \, \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)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

g^2*x^4*e/(-x^2*e^2 + d^2)^(5/2) - 4/3*d^2*g^2*x^2*e^(-1)/(-x^2*e^2 + d^2)^(5/2) + 8/15*d^4*g^2*e^(-3)/(-x^2*e
^2 + d^2)^(5/2) + 2/5*d^3*f*g*e^(-2)/(-x^2*e^2 + d^2)^(5/2) + 3/5*d^2*f^2*e^(-1)/(-x^2*e^2 + d^2)^(5/2) + 1/2*
(3*d*g^2*e^2 + 2*f*g*e^3)*x^3*e^(-2)/(-x^2*e^2 + d^2)^(5/2) - 3/10*(3*d*g^2*e^2 + 2*f*g*e^3)*d^2*x*e^(-4)/(-x^
2*e^2 + d^2)^(5/2) + 1/5*d*f^2*x/(-x^2*e^2 + d^2)^(5/2) + 1/3*(3*d^2*g^2*e + 6*d*f*g*e^2 + f^2*e^3)*x^2*e^(-2)
/(-x^2*e^2 + d^2)^(5/2) - 2/15*(3*d^2*g^2*e + 6*d*f*g*e^2 + f^2*e^3)*d^2*e^(-4)/(-x^2*e^2 + d^2)^(5/2) + 1/10*
(3*d*g^2*e^2 + 2*f*g*e^3)*x*e^(-4)/(-x^2*e^2 + d^2)^(3/2) + 4/15*f^2*x/((-x^2*e^2 + d^2)^(3/2)*d) + 1/5*(d^3*g
^2 + 6*d^2*f*g*e + 3*d*f^2*e^2)*x*e^(-2)/(-x^2*e^2 + d^2)^(5/2) + 1/5*(3*d*g^2*e^2 + 2*f*g*e^3)*x*e^(-4)/(sqrt
(-x^2*e^2 + d^2)*d^2) + 8/15*f^2*x/(sqrt(-x^2*e^2 + d^2)*d^3) - 1/15*(d^3*g^2 + 6*d^2*f*g*e + 3*d*f^2*e^2)*x*e
^(-2)/((-x^2*e^2 + d^2)^(3/2)*d^2) - 2/15*(d^3*g^2 + 6*d^2*f*g*e + 3*d*f^2*e^2)*x*e^(-2)/(sqrt(-x^2*e^2 + d^2)
*d^4)

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Fricas [A]
time = 1.84, size = 266, normalized size = 1.83 \begin {gather*} -\frac {2 \, d^{5} g^{2} - 7 \, f^{2} x^{3} e^{5} + 3 \, {\left (2 \, d f g x^{3} + 7 \, d f^{2} x^{2}\right )} e^{4} - {\left (2 \, d^{2} g^{2} x^{3} + 18 \, d^{2} f g x^{2} + 21 \, d^{2} f^{2} x\right )} e^{3} + {\left (6 \, d^{3} g^{2} x^{2} + 18 \, d^{3} f g x + 7 \, d^{3} f^{2}\right )} e^{2} - 6 \, {\left (d^{4} g^{2} x + d^{4} f g\right )} e + {\left (2 \, d^{4} g^{2} + 2 \, f^{2} x^{2} e^{4} - 6 \, {\left (d f g x^{2} + d f^{2} x\right )} e^{3} + {\left (7 \, d^{2} g^{2} x^{2} + 18 \, d^{2} f g x + 7 \, d^{2} f^{2}\right )} e^{2} - 6 \, {\left (d^{3} g^{2} x + d^{3} f g\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{3} e^{6} - 3 \, d^{4} x^{2} e^{5} + 3 \, d^{5} x e^{4} - d^{6} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(2*d^5*g^2 - 7*f^2*x^3*e^5 + 3*(2*d*f*g*x^3 + 7*d*f^2*x^2)*e^4 - (2*d^2*g^2*x^3 + 18*d^2*f*g*x^2 + 21*d^
2*f^2*x)*e^3 + (6*d^3*g^2*x^2 + 18*d^3*f*g*x + 7*d^3*f^2)*e^2 - 6*(d^4*g^2*x + d^4*f*g)*e + (2*d^4*g^2 + 2*f^2
*x^2*e^4 - 6*(d*f*g*x^2 + d*f^2*x)*e^3 + (7*d^2*g^2*x^2 + 18*d^2*f*g*x + 7*d^2*f^2)*e^2 - 6*(d^3*g^2*x + d^3*f
*g)*e)*sqrt(-x^2*e^2 + d^2))/(d^3*x^3*e^6 - 3*d^4*x^2*e^5 + 3*d^5*x*e^4 - d^6*e^3)

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

[Out]

Integral((d + e*x)**3*(f + g*x)**2/(-(-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. 356 vs. \(2 (134) = 268\).
time = 1.99, size = 356, normalized size = 2.46 \begin {gather*} -\frac {2 \, {\left (\frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} g^{2} e^{\left (-2\right )}}{x} - \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} g^{2} e^{\left (-4\right )}}{x^{2}} - 2 \, d^{2} g^{2} + 6 \, d f g e - \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f g e^{\left (-1\right )}}{x} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f g e^{\left (-3\right )}}{x^{2}} - \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f g e^{\left (-5\right )}}{x^{3}} - 7 \, f^{2} e^{2} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{2} e^{\left (-2\right )}}{x^{2}} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{2} e^{\left (-4\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{2} e^{\left (-6\right )}}{x^{4}} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{2}}{x}\right )} e^{\left (-3\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)^2/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-2/15*(10*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*g^2*e^(-2)/x - 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*g^2*e^(-4)
/x^2 - 2*d^2*g^2 + 6*d*f*g*e - 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d*f*g*e^(-1)/x + 30*(d*e + sqrt(-x^2*e^2 + d^
2)*e)^2*d*f*g*e^(-3)/x^2 - 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d*f*g*e^(-5)/x^3 - 7*f^2*e^2 - 40*(d*e + sqrt(-
x^2*e^2 + d^2)*e)^2*f^2*e^(-2)/x^2 + 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*f^2*e^(-4)/x^3 - 15*(d*e + sqrt(-x^2*
e^2 + d^2)*e)^4*f^2*e^(-6)/x^4 + 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*f^2/x)*e^(-3)/(d^3*((d*e + sqrt(-x^2*e^2 +
d^2)*e)*e^(-2)/x - 1)^5)

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Mupad [B]
time = 2.87, size = 125, normalized size = 0.86 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^4\,g^2-6\,d^3\,e\,f\,g-6\,d^3\,e\,g^2\,x+7\,d^2\,e^2\,f^2+18\,d^2\,e^2\,f\,g\,x+7\,d^2\,e^2\,g^2\,x^2-6\,d\,e^3\,f^2\,x-6\,d\,e^3\,f\,g\,x^2+2\,e^4\,f^2\,x^2\right )}{15\,d^3\,e^3\,{\left (d-e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*(2*d^4*g^2 + 7*d^2*e^2*f^2 + 2*e^4*f^2*x^2 - 6*d^3*e*f*g + 7*d^2*e^2*g^2*x^2 - 6*d*e^3*
f^2*x - 6*d^3*e*g^2*x + 18*d^2*e^2*f*g*x - 6*d*e^3*f*g*x^2))/(15*d^3*e^3*(d - e*x)^3)

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