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

Optimal. Leaf size=138 \[ \frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

[Out]

2/5*(e*x+d)^6/e/(-e^2*x^2+d^2)^(5/2)-14/15*(e*x+d)^4/e/(-e^2*x^2+d^2)^(3/2)-7*d*arctan(e*x/(-e^2*x^2+d^2)^(1/2
))/e+14/3*(e*x+d)^2/e/(-e^2*x^2+d^2)^(1/2)+7*(-e^2*x^2+d^2)^(1/2)/e

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Rubi [A]  time = 0.06, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {669, 641, 217, 203} \[ \frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^6)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (14*(d + e*x)^4)/(15*e*(d^2 - e^2*x^2)^(3/2)) + (14*(d + e*x)^2)
/(3*e*Sqrt[d^2 - e^2*x^2]) + (7*Sqrt[d^2 - e^2*x^2])/e - (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 641

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

Rule 669

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{5} \int \frac {(d+e x)^5}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {7}{3} \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}-7 \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-(7 d) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-(7 d) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 (d+e x)^2}{3 e \sqrt {d^2-e^2 x^2}}+\frac {7 \sqrt {d^2-e^2 x^2}}{e}-\frac {7 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 119, normalized size = 0.86 \[ \frac {(d+e x) \left (\sqrt {1-\frac {e^2 x^2}{d^2}} \left (167 d^3-381 d^2 e x+277 d e^2 x^2-15 e^3 x^3\right )-105 (d-e x)^3 \sin ^{-1}\left (\frac {e x}{d}\right )\right )}{15 e (d-e x)^2 \sqrt {d^2-e^2 x^2} \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(Sqrt[1 - (e^2*x^2)/d^2]*(167*d^3 - 381*d^2*e*x + 277*d*e^2*x^2 - 15*e^3*x^3) - 105*(d - e*x)^3*Arc
Sin[(e*x)/d]))/(15*e*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])

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fricas [A]  time = 1.02, size = 175, normalized size = 1.27 \[ \frac {167 \, d e^{3} x^{3} - 501 \, d^{2} e^{2} x^{2} + 501 \, d^{3} e x - 167 \, d^{4} + 210 \, {\left (d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{3} x^{3} - 277 \, d e^{2} x^{2} + 381 \, d^{2} e x - 167 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(167*d*e^3*x^3 - 501*d^2*e^2*x^2 + 501*d^3*e*x - 167*d^4 + 210*(d*e^3*x^3 - 3*d^2*e^2*x^2 + 3*d^3*e*x - d
^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*e^3*x^3 - 277*d*e^2*x^2 + 381*d^2*e*x - 167*d^3)*sqrt(-e^2
*x^2 + d^2))/(e^4*x^3 - 3*d*e^3*x^2 + 3*d^2*e^2*x - d^3*e)

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giac [A]  time = 0.33, size = 107, normalized size = 0.78 \[ -7 \, d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\relax (d) - \frac {{\left (167 \, d^{6} e^{\left (-1\right )} + {\left (120 \, d^{5} - {\left (365 \, d^{4} e + {\left (160 \, d^{3} e^{2} - {\left (405 \, d^{2} e^{3} - {\left (15 \, x e^{5} - 232 \, d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*d*arcsin(x*e/d)*e^(-1)*sgn(d) - 1/15*(167*d^6*e^(-1) + (120*d^5 - (365*d^4*e + (160*d^3*e^2 - (405*d^2*e^3
- (15*x*e^5 - 232*d*e^4)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3

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maple [B]  time = 0.12, size = 253, normalized size = 1.83 \[ -\frac {e^{5} x^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d \,e^{4} x^{5}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {27 d^{2} e^{3} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {35 d^{3} e^{2} x^{3}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {73 d^{4} e \,x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {61 d^{5} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {7 d \,e^{2} x^{3}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {167 d^{6}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {71 d^{3} x}{30 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {176 d x}{15 \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {7 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^5*x^6/(-e^2*x^2+d^2)^(5/2)+27*e^3*d^2*x^4/(-e^2*x^2+d^2)^(5/2)-73/3*e*d^4*x^2/(-e^2*x^2+d^2)^(5/2)+167/15*d
^6/e/(-e^2*x^2+d^2)^(5/2)+7/5*d*e^4*x^5/(-e^2*x^2+d^2)^(5/2)-7/3/(-e^2*x^2+d^2)^(3/2)*d*e^2*x^3+176/15/(-e^2*x
^2+d^2)^(1/2)*d*x-7/(e^2)^(1/2)*d*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)+35/2*d^3*e^2*x^3/(-e^2*x^2+d^2)^(
5/2)-61/10*d^5*x/(-e^2*x^2+d^2)^(5/2)+71/30/(-e^2*x^2+d^2)^(3/2)*d^3*x

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maxima [B]  time = 3.16, size = 318, normalized size = 2.30 \[ \frac {7}{15} \, d e^{6} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {7}{3} \, d e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {27 \, d^{2} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {35 \, d^{3} e^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {73 \, d^{4} e x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {61 \, d^{5} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {167 \, d^{6}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {127 \, d^{3} x}{30 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {22 \, d x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}}} - \frac {7 \, d \arcsin \left (\frac {e x}{d}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/15*d*e^6*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2
+ d^2)^(5/2)*e^6)) - e^5*x^6/(-e^2*x^2 + d^2)^(5/2) - 7/3*d*e^4*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2
/((-e^2*x^2 + d^2)^(3/2)*e^4)) + 27*d^2*e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 35/2*d^3*e^2*x^3/(-e^2*x^2 + d^2)^(5/
2) - 73/3*d^4*e*x^2/(-e^2*x^2 + d^2)^(5/2) - 61/10*d^5*x/(-e^2*x^2 + d^2)^(5/2) + 167/15*d^6/((-e^2*x^2 + d^2)
^(5/2)*e) + 127/30*d^3*x/(-e^2*x^2 + d^2)^(3/2) + 22/15*d*x/sqrt(-e^2*x^2 + d^2) - 7*d*arcsin(e*x/d)/e

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x\right )}^7}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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