### 3.851 $$\int \frac{(d+e x)^4}{(d^2-e^2 x^2)^{7/2}} \, dx$$

Optimal. Leaf size=67 $\frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}}$

[Out]

(d + e*x)^4/(3*d*e*(d^2 - e^2*x^2)^(5/2)) - (d + e*x)^5/(15*d^2*e*(d^2 - e^2*x^2)^(5/2))

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Rubi [A]  time = 0.0227085, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {659, 651} $\frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d + e*x)^4/(3*d*e*(d^2 - e^2*x^2)^(5/2)) - (d + e*x)^5/(15*d^2*e*(d^2 - e^2*x^2)^(5/2))

Rule 659

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

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0687201, size = 49, normalized size = 0.73 $\frac{(4 d-e x) (d+e x)^2}{15 d^2 e (d-e x)^2 \sqrt{d^2-e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((4*d - e*x)*(d + e*x)^2)/(15*d^2*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])

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Maple [A]  time = 0.043, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( ex+d \right ) ^{5} \left ( -ex+d \right ) \left ( -ex+4\,d \right ) }{15\,{d}^{2}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.17327, size = 166, normalized size = 2.48 \begin{align*} \frac{e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{11 \, d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d^{3}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} - \frac{x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) + 4/3*d*e*x^2/(-e^2*x^2 + d^2)^(5/2) + 11/10*d^2*x/(-e^2*x^2 + d^2)^(5/2) +
4/15*d^3/((-e^2*x^2 + d^2)^(5/2)*e) - 1/30*x/(-e^2*x^2 + d^2)^(3/2) - 1/15*x/(sqrt(-e^2*x^2 + d^2)*d^2)

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Fricas [A]  time = 2.08619, size = 212, normalized size = 3.16 \begin{align*} \frac{4 \, e^{3} x^{3} - 12 \, d e^{2} x^{2} + 12 \, d^{2} e x - 4 \, d^{3} +{\left (e^{2} x^{2} - 3 \, d e x - 4 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{4} x^{3} - 3 \, d^{3} e^{3} x^{2} + 3 \, d^{4} e^{2} x - d^{5} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(4*e^3*x^3 - 12*d*e^2*x^2 + 12*d^2*e*x - 4*d^3 + (e^2*x^2 - 3*d*e*x - 4*d^2)*sqrt(-e^2*x^2 + d^2))/(d^2*e
^4*x^3 - 3*d^3*e^3*x^2 + 3*d^4*e^2*x - d^5*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**4/(-(-d + e*x)*(d + e*x))**(7/2), x)

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Giac [A]  time = 1.34625, size = 95, normalized size = 1.42 \begin{align*} -\frac{{\left (4 \, d^{3} e^{\left (-1\right )} +{\left (15 \, d^{2} -{\left (x{\left (\frac{x^{2} e^{4}}{d^{2}} - 10 \, e^{2}\right )} - 20 \, d e\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-1/15*(4*d^3*e^(-1) + (15*d^2 - (x*(x^2*e^4/d^2 - 10*e^2) - 20*d*e)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)
^3