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

Optimal. Leaf size=160 \[ -\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}-\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}} \]

[Out]

(-256*d^3*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e*Sqrt[d + e*x]) - (64*d^2*Sqrt[d + e*x
]*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e) - (24*d*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x
^2])/(35*c*e) - (2*(d + e*x)^(5/2)*Sqrt[c*d^2 - c*e^2*x^2])/(7*c*e)

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Rubi [A]  time = 0.238602, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}-\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-256*d^3*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e*Sqrt[d + e*x]) - (64*d^2*Sqrt[d + e*x
]*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e) - (24*d*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x
^2])/(35*c*e) - (2*(d + e*x)^(5/2)*Sqrt[c*d^2 - c*e^2*x^2])/(7*c*e)

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Rubi in Sympy [A]  time = 21.628, size = 138, normalized size = 0.86 \[ - \frac{256 d^{3} \sqrt{c d^{2} - c e^{2} x^{2}}}{35 c e \sqrt{d + e x}} - \frac{64 d^{2} \sqrt{d + e x} \sqrt{c d^{2} - c e^{2} x^{2}}}{35 c e} - \frac{24 d \left (d + e x\right )^{\frac{3}{2}} \sqrt{c d^{2} - c e^{2} x^{2}}}{35 c e} - \frac{2 \left (d + e x\right )^{\frac{5}{2}} \sqrt{c d^{2} - c e^{2} x^{2}}}{7 c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

-256*d**3*sqrt(c*d**2 - c*e**2*x**2)/(35*c*e*sqrt(d + e*x)) - 64*d**2*sqrt(d + e
*x)*sqrt(c*d**2 - c*e**2*x**2)/(35*c*e) - 24*d*(d + e*x)**(3/2)*sqrt(c*d**2 - c*
e**2*x**2)/(35*c*e) - 2*(d + e*x)**(5/2)*sqrt(c*d**2 - c*e**2*x**2)/(7*c*e)

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Mathematica [A]  time = 0.0642337, size = 70, normalized size = 0.44 \[ -\frac{2 (d-e x) \sqrt{d+e x} \left (177 d^3+71 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )}{35 e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d - e*x)*Sqrt[d + e*x]*(177*d^3 + 71*d^2*e*x + 27*d*e^2*x^2 + 5*e^3*x^3))/(
35*e*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.01, size = 66, normalized size = 0.4 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 5\,{e}^{3}{x}^{3}+27\,d{e}^{2}{x}^{2}+71\,{d}^{2}xe+177\,{d}^{3} \right ) }{35\,e}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(1/2),x)

[Out]

-2/35*(-e*x+d)*(5*e^3*x^3+27*d*e^2*x^2+71*d^2*e*x+177*d^3)*(e*x+d)^(1/2)/e/(-c*e
^2*x^2+c*d^2)^(1/2)

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Maxima [A]  time = 0.720183, size = 77, normalized size = 0.48 \[ \frac{2 \,{\left (5 \, e^{4} x^{4} + 22 \, d e^{3} x^{3} + 44 \, d^{2} e^{2} x^{2} + 106 \, d^{3} e x - 177 \, d^{4}\right )}}{35 \, \sqrt{-e x + d} \sqrt{c} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(-c*e^2*x^2 + c*d^2),x, algorithm="maxima")

[Out]

2/35*(5*e^4*x^4 + 22*d*e^3*x^3 + 44*d^2*e^2*x^2 + 106*d^3*e*x - 177*d^4)/(sqrt(-
e*x + d)*sqrt(c)*e)

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Fricas [A]  time = 0.224097, size = 109, normalized size = 0.68 \[ \frac{2 \,{\left (5 \, e^{5} x^{5} + 27 \, d e^{4} x^{4} + 66 \, d^{2} e^{3} x^{3} + 150 \, d^{3} e^{2} x^{2} - 71 \, d^{4} e x - 177 \, d^{5}\right )}}{35 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(-c*e^2*x^2 + c*d^2),x, algorithm="fricas")

[Out]

2/35*(5*e^5*x^5 + 27*d*e^4*x^4 + 66*d^2*e^3*x^3 + 150*d^3*e^2*x^2 - 71*d^4*e*x -
177*d^5)/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(7/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{-c e^{2} x^{2} + c d^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(-c*e^2*x^2 + c*d^2),x, algorithm="giac")

[Out]

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