3.7.91 \(\int \frac {(d+e x)^5}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac {(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {651} \begin {gather*} \frac {(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d + e*x)^5/(5*d*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]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 41, normalized size = 1.24 \begin {gather*} \frac {(d+e x)^3}{5 d e (d-e x)^2 \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d + e*x)^3/(5*d*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2])

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IntegrateAlgebraic [A]  time = 0.44, size = 50, normalized size = 1.52 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (d^2+2 d e x+e^2 x^2\right )}{5 d e (d-e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(d^2 + 2*d*e*x + e^2*x^2))/(5*d*e*(d - e*x)^3)

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fricas [B]  time = 0.41, size = 100, normalized size = 3.03 \begin {gather*} \frac {e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3} - {\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d e^{4} x^{3} - 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x - d^{4} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.30, size = 76, normalized size = 2.30 \begin {gather*} -\frac {{\left (d^{4} e^{\left (-1\right )} + {\left (5 \, d^{3} + {\left (10 \, d^{2} e + {\left (x {\left (\frac {x e^{4}}{d} + 5 \, e^{3}\right )} + 10 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\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)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 36, normalized size = 1.09 \begin {gather*} \frac {\left (e x +d \right )^{6} \left (-e x +d \right )}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/5*(e*x+d)^6*(-e*x+d)/d/e/(-e^2*x^2+d^2)^(7/2)

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maxima [B]  time = 1.45, size = 148, normalized size = 4.48 \begin {gather*} \frac {e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {5 \, d e^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {7 \, d^{3} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d^{4}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 5/2*d*e^2*x^3/(-e^2*x^2 + d^2)^(5/2) + 2*d^2*e*x^2/(-e^2*x^2 + d^2)^(5/2) + 7
/10*d^3*x/(-e^2*x^2 + d^2)^(5/2) + 1/5*d^4/((-e^2*x^2 + d^2)^(5/2)*e) + 1/10*d*x/(-e^2*x^2 + d^2)^(3/2) + 1/5*
x/(sqrt(-e^2*x^2 + d^2)*d)

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mupad [B]  time = 0.87, size = 37, normalized size = 1.12 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,{\left (d+e\,x\right )}^2}{5\,d\,e\,{\left (d-e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x)

[Out]

((d^2 - e^2*x^2)^(1/2)*(d + e*x)^2)/(5*d*e*(d - e*x)^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 )^{5}}{\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)**5/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**5/(-(-d + e*x)*(d + e*x))**(7/2), x)

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