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]

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

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Rubi [A]  time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.98, size = 104, normalized size = 1.55 \[ \frac {4 \, e^{3} x^{3} - 12 \, d e^{2} x^{2} + 12 \, d^{2} e x - 4 \, d^{3} + {\left (e^{2} x^{2} - 3 \, d e x - 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{4} x^{3} - 3 \, d^{3} e^{3} x^{2} + 3 \, d^{4} e^{2} x - d^{5} e\right )}} \]

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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giac [A]  time = 0.30, size = 70, normalized size = 1.04 \[ -\frac {{\left (4 \, d^{3} e^{\left (-1\right )} + {\left (15 \, d^{2} - {\left (x {\left (\frac {x^{2} e^{4}}{d^{2}} - 10 \, e^{2}\right )} - 20 \, d e\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

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

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maple [A]  time = 0.05, size = 44, normalized size = 0.66 \[ \frac {\left (e x +d \right )^{5} \left (-e x +d \right ) \left (-e x +4 d \right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2} e} \]

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 = 1.39, size = 123, normalized size = 1.84 \[ \frac {e^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {11 \, d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d^{3}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {x}{30 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}} \]

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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mupad [B]  time = 0.68, size = 49, normalized size = 0.73 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (4\,d^2+3\,d\,e\,x-e^2\,x^2\right )}{15\,d^2\,e\,{\left (d-e\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]

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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