\(\int \sqrt {d+e x} (c d^2-c e^2 x^2)^{3/2} \, dx\) [871]

Optimal result

Integrand size = 29, antiderivative size = 119 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}} \]

[Out]

-64/315*d^2*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(5/2)-16/63*d*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(3/2)-2/9*
(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}} \]

[In]

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

[Out]

(-64*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(315*c*e*(d + e*x)^(5/2)) - (16*d*(c*d^2 - c*e^2*x^2)^(5/2))/(63*c*e*(d +
e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(9*c*e*Sqrt[d + e*x])

Rule 663

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 671

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*} \text {integral}& = -\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}+\frac {1}{9} (8 d) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx \\ & = -\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}+\frac {1}{63} \left (32 d^2\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx \\ & = -\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.52 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 c (d-e x)^2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (107 d^2+110 d e x+35 e^2 x^2\right )}{315 e \sqrt {d+e x}} \]

[In]

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

[Out]

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(107*d^2 + 110*d*e*x + 35*e^2*x^2))/(315*e*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.46

method	result	size
gosper	\(-\frac {2 \left (-e x +d \right ) \left (35 x^{2} e^{2}+110 d e x +107 d^{2}\right ) \left (-c \,x^{2} e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{315 e \left (e x +d \right )^{\frac {3}{2}}}\)	\(55\)
default	\(-\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, c \left (-e x +d \right )^{2} \left (35 x^{2} e^{2}+110 d e x +107 d^{2}\right )}{315 \sqrt {e x +d}\, e}\)	\(57\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2} \left (35 e^{4} x^{4}+40 d \,e^{3} x^{3}-78 d^{2} e^{2} x^{2}-104 d^{3} e x +107 d^{4}\right ) \left (-e x +d \right )}{315 \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(118\)

[In]

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

[Out]

-2/315*(-e*x+d)*(35*e^2*x^2+110*d*e*x+107*d^2)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (35 \, c e^{4} x^{4} + 40 \, c d e^{3} x^{3} - 78 \, c d^{2} e^{2} x^{2} - 104 \, c d^{3} e x + 107 \, c d^{4}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \]

[In]

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

[Out]

-2/315*(35*c*e^4*x^4 + 40*c*d*e^3*x^3 - 78*c*d^2*e^2*x^2 - 104*c*d^3*e*x + 107*c*d^4)*sqrt(-c*e^2*x^2 + c*d^2)
*sqrt(e*x + d)/(e^2*x + d*e)

Sympy [F]

\[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\int \left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (35 \, c^{\frac {3}{2}} e^{4} x^{4} + 40 \, c^{\frac {3}{2}} d e^{3} x^{3} - 78 \, c^{\frac {3}{2}} d^{2} e^{2} x^{2} - 104 \, c^{\frac {3}{2}} d^{3} e x + 107 \, c^{\frac {3}{2}} d^{4}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{315 \, {\left (e^{2} x + d e\right )}} \]

[In]

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

[Out]

-2/315*(35*c^(3/2)*e^4*x^4 + 40*c^(3/2)*d*e^3*x^3 - 78*c^(3/2)*d^2*e^2*x^2 - 104*c^(3/2)*d^3*e*x + 107*c^(3/2)
*d^4)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (101) = 202\).

Time = 0.29 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.30 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (105 \, {\left (2 \, \sqrt {2} \sqrt {c d} d - \frac {{\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c}\right )} c d^{3} - 3 \, {\left (\frac {22 \, \sqrt {2} \sqrt {c d} d^{3}}{e^{2}} - \frac {35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2} d^{2} - 42 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - 15 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{3} e^{2}}\right )} c d e^{2} + {\left (\frac {26 \, \sqrt {2} \sqrt {c d} d^{4}}{e^{3}} + \frac {105 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{3} d^{3} - 189 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{2} d^{2} - 135 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - 35 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{4} e^{3}}\right )} c e^{3} - 21 \, {\left (2 \, \sqrt {2} \sqrt {c d} d^{2} + \frac {5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{2}}\right )} c d^{2}\right )}}{315 \, e} \]

[In]

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

[Out]

2/315*(105*(2*sqrt(2)*sqrt(c*d)*d - (-(e*x + d)*c + 2*c*d)^(3/2)/c)*c*d^3 - 3*(22*sqrt(2)*sqrt(c*d)*d^3/e^2 -
(35*(-(e*x + d)*c + 2*c*d)^(3/2)*c^2*d^2 - 42*((e*x + d)*c - 2*c*d)^2*sqrt(-(e*x + d)*c + 2*c*d)*c*d - 15*((e*
x + d)*c - 2*c*d)^3*sqrt(-(e*x + d)*c + 2*c*d))/(c^3*e^2))*c*d*e^2 + (26*sqrt(2)*sqrt(c*d)*d^4/e^3 + (105*(-(e
*x + d)*c + 2*c*d)^(3/2)*c^3*d^3 - 189*((e*x + d)*c - 2*c*d)^2*sqrt(-(e*x + d)*c + 2*c*d)*c^2*d^2 - 135*((e*x
+ d)*c - 2*c*d)^3*sqrt(-(e*x + d)*c + 2*c*d)*c*d - 35*((e*x + d)*c - 2*c*d)^4*sqrt(-(e*x + d)*c + 2*c*d))/(c^4
*e^3))*c*e^3 - 21*(2*sqrt(2)*sqrt(c*d)*d^2 + (5*(-(e*x + d)*c + 2*c*d)^(3/2)*c*d - 3*((e*x + d)*c - 2*c*d)^2*s
qrt(-(e*x + d)*c + 2*c*d))/c^2)*c*d^2)/e

Mupad [B] (verification not implemented)

Time = 10.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {214\,c\,d^4\,\sqrt {d+e\,x}}{315\,e^2}-\frac {52\,c\,d^2\,x^2\,\sqrt {d+e\,x}}{105}+\frac {2\,c\,e^2\,x^4\,\sqrt {d+e\,x}}{9}-\frac {208\,c\,d^3\,x\,\sqrt {d+e\,x}}{315\,e}+\frac {16\,c\,d\,e\,x^3\,\sqrt {d+e\,x}}{63}\right )}{x+\frac {d}{e}} \]

[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)*((214*c*d^4*(d + e*x)^(1/2))/(315*e^2) - (52*c*d^2*x^2*(d + e*x)^(1/2))/105 + (2*c
*e^2*x^4*(d + e*x)^(1/2))/9 - (208*c*d^3*x*(d + e*x)^(1/2))/(315*e) + (16*c*d*e*x^3*(d + e*x)^(1/2))/63))/(x +
d/e)