3.9.50 \(\int \frac {(d+e x)^5}{(d^2-e^2 x^2)^{7/2}} \, dx\) [850]

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

[Out]

1/5*(e*x+d)^5/d/e/(-e^2*x^2+d^2)^(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 = {665} \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 665

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 = 0.38, size = 41, normalized size = 1.24 \begin {gather*} \frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{5 d e (d-e x)^3} \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)^2*Sqrt[d^2 - e^2*x^2])/(5*d*e*(d - e*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(462\) vs. \(2(29)=58\).
time = 0.47, size = 463, normalized size = 14.03

method	result	size
gosper	\(\frac {\left (e x +d \right )^{6} \left (-e x +d \right )}{5 d e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(36\)
trager	\(\frac {\left (e^{2} x^{2}+2 d x e +d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d \left (-e x +d \right )^{3} e}\)	\(47\)
default	\(e^{5} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+5 d \,e^{4} \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+10 d^{2} e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+10 d^{3} e^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {d^{4}}{e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{5} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\)	\(463\)

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,method=_RETURNVERBOSE)

[Out]

e^5*(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
)))+5*d*e^4*(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)))))+10*d^2*e^3*
(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))+10*d^3*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))))+d^4/e/(-e^2*x^2+d^2)^(5/2)+d^5*(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. 139 vs. \(2 (28) = 56\).
time = 0.28, size = 139, normalized size = 4.21 \begin {gather*} \frac {x^{4} e^{3}}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {5 \, d x^{3} e^{2}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d^{2} x^{2} e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d^{4} e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {7 \, d^{3} x}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d x}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {x}{5 \, \sqrt {-x^{2} e^{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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (28) = 56\).
time = 2.24, size = 96, normalized size = 2.91 \begin {gather*} \frac {x^{3} e^{3} - 3 \, d x^{2} e^{2} + 3 \, d^{2} x e - d^{3} - {\left (x^{2} e^{2} + 2 \, d x e + d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (d x^{3} e^{4} - 3 \, d^{2} x^{2} e^{3} + 3 \, d^{3} x e^{2} - 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*(x^3*e^3 - 3*d*x^2*e^2 + 3*d^2*x*e - d^3 - (x^2*e^2 + 2*d*x*e + d^2)*sqrt(-x^2*e^2 + d^2))/(d*x^3*e^4 - 3*
d^2*x^2*e^3 + 3*d^3*x*e^2 - d^4*e)

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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (28) = 56\).
time = 3.04, size = 100, normalized size = 3.03 \begin {gather*} \frac {2 \, {\left (\frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + 1\right )} e^{\left (-1\right )}}{5 \, d {\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)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

2/5*(10*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-4)/x^2 + 5*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-8)/x^4 + 1)*e^(-1
)/(d*((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 1)^5)

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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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