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

Optimal. Leaf size=78 \[ -\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}} \]

[Out]

-8/35*d*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(5/2)-2/7*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 78, 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} \[ -\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(35*c*e*(d + e*x)^(5/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(7*c*e*(d + e*x)^(3/
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 {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx &=-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}+\frac {1}{7} (4 d) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\\ &=-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 51, normalized size = 0.65 \[ -\frac {2 c (d-e x)^2 (9 d+5 e x) \sqrt {c \left (d^2-e^2 x^2\right )}}{35 e \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(d - e*x)^2*(9*d + 5*e*x)*Sqrt[c*(d^2 - e^2*x^2)])/(35*e*Sqrt[d + e*x])

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 71, normalized size = 0.91 \[ -\frac {2 \, {\left (5 \, c e^{3} x^{3} - c d e^{2} x^{2} - 13 \, c d^{2} e x + 9 \, c d^{3}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{35 \, {\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/35*(5*c*e^3*x^3 - c*d*e^2*x^2 - 13*c*d^2*e*x + 9*c*d^3)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e
)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 44, normalized size = 0.56 \[ -\frac {2 \left (-e x +d \right ) \left (5 e x +9 d \right ) \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{\frac {3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(-e*x+d)*(5*e*x+9*d)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.50, size = 55, normalized size = 0.71 \[ -\frac {2 \, {\left (5 \, c^{\frac {3}{2}} e^{3} x^{3} - c^{\frac {3}{2}} d e^{2} x^{2} - 13 \, c^{\frac {3}{2}} d^{2} e x + 9 \, c^{\frac {3}{2}} d^{3}\right )} \sqrt {-e x + d}}{35 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-2/35*(5*c^(3/2)*e^3*x^3 - c^(3/2)*d*e^2*x^2 - 13*c^(3/2)*d^2*e*x + 9*c^(3/2)*d^3)*sqrt(-e*x + d)/e

________________________________________________________________________________________

mupad [B]  time = 0.59, size = 60, normalized size = 0.77 \[ -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {18\,c\,d^3}{35\,e}+\frac {2\,c\,e^2\,x^3}{7}-\frac {26\,c\,d^2\,x}{35}-\frac {2\,c\,d\,e\,x^2}{35}\right )}{\sqrt {d+e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((18*c*d^3)/(35*e) + (2*c*e^2*x^3)/7 - (26*c*d^2*x)/35 - (2*c*d*e*x^2)/35))/(d + e
*x)^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

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

________________________________________________________________________________________