3.888 \(\int \frac {(d+e x)^{5/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}} \]

[Out]

-2*(e*x+d)^(3/2)/c/e/(-c*e^2*x^2+c*d^2)^(1/2)+8*d*(e*x+d)^(1/2)/c/e/(-c*e^2*x^2+c*d^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {657, 649} \[ \frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

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

Rule 649

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] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rule 657

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*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*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] && IGtQ[Simplify[m + p]
, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}}+(4 d) \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 43, normalized size = 0.58 \[ \frac {2 (3 d-e x) \sqrt {d+e x}}{c e \sqrt {c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.99, size = 55, normalized size = 0.74 \[ \frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} {\left (e x - 3 \, d\right )}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*(e*x - 3*d)/(c^2*e^3*x^2 - c^2*d^2*e)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 44, normalized size = 0.59 \[ \frac {2 \left (-e x +d \right ) \left (-e x +3 d \right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

2*(-e*x+d)*(-e*x+3*d)*(e*x+d)^(3/2)/e/(-c*e^2*x^2+c*d^2)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.52, size = 23, normalized size = 0.31 \[ -\frac {2 \, {\left (e x - 3 \, d\right )}}{\sqrt {-e x + d} c^{\frac {3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2*(e*x - 3*d)/(sqrt(-e*x + d)*c^(3/2)*e)

________________________________________________________________________________________

mupad [B]  time = 0.60, size = 66, normalized size = 0.89 \[ -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {6\,d\,\sqrt {d+e\,x}}{c^2\,e^3}-\frac {2\,x\,\sqrt {d+e\,x}}{c^2\,e^2}\right )}{x^2-\frac {d^2}{e^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x)

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((6*d*(d + e*x)^(1/2))/(c^2*e^3) - (2*x*(d + e*x)^(1/2))/(c^2*e^2)))/(x^2 - d^2/e^
2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((d + e*x)**(5/2)/(-c*(-d + e*x)*(d + e*x))**(3/2), x)

________________________________________________________________________________________