3.9.51 \(\int \frac {(d+e x)^4}{(d^2-e^2 x^2)^{7/2}} \, dx\) [851]

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 = {673, 665} \begin {gather*} \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 {gather*}

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 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]

Rule 673

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 = 0.38, size = 53, normalized size = 0.79 \begin {gather*} \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 (d-e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(4*d^2 + 3*d*e*x - e^2*x^2))/(15*d^2*e*(d - e*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(380\) vs. \(2(59)=118\).
time = 0.46, size = 381, normalized size = 5.69

method	result	size
gosper	\(\frac {\left (e x +d \right )^{5} \left (-e x +d \right ) \left (-e x +4 d \right )}{15 d^{2} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(44\)
trager	\(\frac {\left (-e^{2} x^{2}+3 d x e +4 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} \left (-e x +d \right )^{3} e}\)	\(50\)
default	\(e^{4} \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+4 d \,e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+6 d^{2} e^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {4 d^{3}}{5 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{4} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\)	\(381\)

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,method=_RETURNVERBOSE)

[Out]

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)))))+4*d*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))+6*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))))+4/
5*d^3/e/(-e^2*x^2+d^2)^(5/2)+d^4*(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. 116 vs. \(2 (57) = 114\).
time = 0.29, size = 116, normalized size = 1.73 \begin {gather*} \frac {x^{3} e^{2}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d x^{2} e}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d^{3} e^{\left (-1\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {11 \, d^{2} x}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {x}{30 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} \end {gather*}

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

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Fricas [A]
time = 2.44, size = 100, normalized size = 1.49 \begin {gather*} \frac {4 \, x^{3} e^{3} - 12 \, d x^{2} e^{2} + 12 \, d^{2} x e - 4 \, d^{3} + {\left (x^{2} e^{2} - 3 \, d x e - 4 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{2} x^{3} e^{4} - 3 \, d^{3} x^{2} e^{3} + 3 \, d^{4} x e^{2} - d^{5} e\right )}} \end {gather*}

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*x^3*e^3 - 12*d*x^2*e^2 + 12*d^2*x*e - 4*d^3 + (x^2*e^2 - 3*d*x*e - 4*d^2)*sqrt(-x^2*e^2 + d^2))/(d^2*x
^3*e^4 - 3*d^3*x^2*e^3 + 3*d^4*x*e^2 - d^5*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 )^{4}}{\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)**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 [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (57) = 114\).
time = 4.13, size = 158, normalized size = 2.36 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - \frac {25 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} - 4\right )} e^{\left (-1\right )}}{15 \, d^{2} {\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)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-2/15*(5*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 25*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^(-4)/x^2 + 15*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^3*e^(-6)/x^3 - 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-8)/x^4 - 4)*e^(-1)/(d^2*((d*e +
sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 1)^5)

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Mupad [B]
time = 0.68, size = 49, normalized size = 0.73 \begin {gather*} \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} \end {gather*}

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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