\(\int (d+e x)^{5/2} (c d^2-c e^2 x^2)^{3/2} \, dx\) [869]

Optimal result

Integrand size = 29, antiderivative size = 201 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e} \]

[Out]

-4096/15015*d^4*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(5/2)-1024/3003*d^3*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^
(3/2)-2/13*(e*x+d)^(3/2)*(-c*e^2*x^2+c*d^2)^(5/2)/c/e-128/429*d^2*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(1/2)-3
2/143*d*(-c*e^2*x^2+c*d^2)^(5/2)*(e*x+d)^(1/2)/c/e

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}-\frac {4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}} \]

[In]

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

[Out]

(-4096*d^4*(c*d^2 - c*e^2*x^2)^(5/2))/(15015*c*e*(d + e*x)^(5/2)) - (1024*d^3*(c*d^2 - c*e^2*x^2)^(5/2))/(3003
*c*e*(d + e*x)^(3/2)) - (128*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(429*c*e*Sqrt[d + e*x]) - (32*d*Sqrt[d + e*x]*(c*d
^2 - c*e^2*x^2)^(5/2))/(143*c*e) - (2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(5/2))/(13*c*e)

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 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {1}{13} (16 d) \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx \\ & = -\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {1}{143} \left (192 d^2\right ) \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx \\ & = -\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {1}{429} \left (512 d^3\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx \\ & = -\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {\left (2048 d^4\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003} \\ & = -\frac {4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.42 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 c (d-e x)^2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (9683 d^4+16700 d^3 e x+14210 d^2 e^2 x^2+6300 d e^3 x^3+1155 e^4 x^4\right )}{15015 e \sqrt {d+e x}} \]

[In]

Integrate[(d + e*x)^(5/2)*(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)]*(9683*d^4 + 16700*d^3*e*x + 14210*d^2*e^2*x^2 + 6300*d*e^3*x^3 + 115
5*e^4*x^4))/(15015*e*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.38

method	result	size
gosper	\(-\frac {2 \left (-e x +d \right ) \left (1155 e^{4} x^{4}+6300 d \,e^{3} x^{3}+14210 d^{2} e^{2} x^{2}+16700 d^{3} e x +9683 d^{4}\right ) \left (-c \,x^{2} e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{15015 e \left (e x +d \right )^{\frac {3}{2}}}\)	\(77\)
default	\(-\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, c \left (-e x +d \right )^{2} \left (1155 e^{4} x^{4}+6300 d \,e^{3} x^{3}+14210 d^{2} e^{2} x^{2}+16700 d^{3} e x +9683 d^{4}\right )}{15015 \sqrt {e x +d}\, e}\)	\(79\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2} \left (1155 e^{6} x^{6}+3990 d \,e^{5} x^{5}+2765 d^{2} e^{4} x^{4}-5420 x^{3} d^{3} e^{3}-9507 d^{4} e^{2} x^{2}-2666 d^{5} e x +9683 d^{6}\right ) \left (-e x +d \right )}{15015 \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(140\)

[In]

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

[Out]

-2/15015*(-e*x+d)*(1155*e^4*x^4+6300*d*e^3*x^3+14210*d^2*e^2*x^2+16700*d^3*e*x+9683*d^4)*(-c*e^2*x^2+c*d^2)^(3
/2)/e/(e*x+d)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.53 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (1155 \, c e^{6} x^{6} + 3990 \, c d e^{5} x^{5} + 2765 \, c d^{2} e^{4} x^{4} - 5420 \, c d^{3} e^{3} x^{3} - 9507 \, c d^{4} e^{2} x^{2} - 2666 \, c d^{5} e x + 9683 \, c d^{6}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{15015 \, {\left (e^{2} x + d e\right )}} \]

[In]

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

[Out]

-2/15015*(1155*c*e^6*x^6 + 3990*c*d*e^5*x^5 + 2765*c*d^2*e^4*x^4 - 5420*c*d^3*e^3*x^3 - 9507*c*d^4*e^2*x^2 - 2
666*c*d^5*e*x + 9683*c*d^6)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e)

Sympy [F]

\[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\int \left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.55 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (1155 \, c^{\frac {3}{2}} e^{6} x^{6} + 3990 \, c^{\frac {3}{2}} d e^{5} x^{5} + 2765 \, c^{\frac {3}{2}} d^{2} e^{4} x^{4} - 5420 \, c^{\frac {3}{2}} d^{3} e^{3} x^{3} - 9507 \, c^{\frac {3}{2}} d^{4} e^{2} x^{2} - 2666 \, c^{\frac {3}{2}} d^{5} e x + 9683 \, c^{\frac {3}{2}} d^{6}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{15015 \, {\left (e^{2} x + d e\right )}} \]

[In]

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

[Out]

-2/15015*(1155*c^(3/2)*e^6*x^6 + 3990*c^(3/2)*d*e^5*x^5 + 2765*c^(3/2)*d^2*e^4*x^4 - 5420*c^(3/2)*d^3*e^3*x^3
- 9507*c^(3/2)*d^4*e^2*x^2 - 2666*c^(3/2)*d^5*e*x + 9683*c^(3/2)*d^6)*(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. 822 vs. \(2 (171) = 342\).

Time = 0.32 (sec) , antiderivative size = 822, normalized size of antiderivative = 4.09 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (15015 \, {\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^{5} + 858 \, {\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^{3} e^{2} + 286 \, {\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 d^{2} e^{3} - 39 \, {\left (\frac {422 \, \sqrt {2} \sqrt {c d} d^{5}}{e^{4}} - \frac {1155 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{4} d^{4} - 2772 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{3} d^{3} - 2970 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{2} d^{2} - 1540 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - 315 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{5} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{5} e^{4}}\right )} c d e^{4} + 5 \, {\left (\frac {542 \, \sqrt {2} \sqrt {c d} d^{6}}{e^{5}} + \frac {3003 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{5} d^{5} - 9009 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{4} d^{4} - 12870 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{3} d^{3} - 10010 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{2} d^{2} - 4095 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{5} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - 693 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{6} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{6} e^{5}}\right )} c e^{5} - 9009 \, {\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^{4}\right )}}{45045 \, e} \]

[In]

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

[Out]

2/45045*(15015*(2*sqrt(2)*sqrt(c*d)*d - (-(e*x + d)*c + 2*c*d)^(3/2)/c)*c*d^5 + 858*(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 - 1
5*((e*x + d)*c - 2*c*d)^3*sqrt(-(e*x + d)*c + 2*c*d))/(c^3*e^2))*c*d^3*e^2 + 286*(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*d^2*e^3 - 39*(422*sqrt(2)*sqrt(c*d)*d^5/e^4 - (1155*(-(e*x + d)*c + 2*c*d)^(3/2)*c^4*d^4
- 2772*((e*x + d)*c - 2*c*d)^2*sqrt(-(e*x + d)*c + 2*c*d)*c^3*d^3 - 2970*((e*x + d)*c - 2*c*d)^3*sqrt(-(e*x +
d)*c + 2*c*d)*c^2*d^2 - 1540*((e*x + d)*c - 2*c*d)^4*sqrt(-(e*x + d)*c + 2*c*d)*c*d - 315*((e*x + d)*c - 2*c*d
)^5*sqrt(-(e*x + d)*c + 2*c*d))/(c^5*e^4))*c*d*e^4 + 5*(542*sqrt(2)*sqrt(c*d)*d^6/e^5 + (3003*(-(e*x + d)*c +
2*c*d)^(3/2)*c^5*d^5 - 9009*((e*x + d)*c - 2*c*d)^2*sqrt(-(e*x + d)*c + 2*c*d)*c^4*d^4 - 12870*((e*x + d)*c -
2*c*d)^3*sqrt(-(e*x + d)*c + 2*c*d)*c^3*d^3 - 10010*((e*x + d)*c - 2*c*d)^4*sqrt(-(e*x + d)*c + 2*c*d)*c^2*d^2
- 4095*((e*x + d)*c - 2*c*d)^5*sqrt(-(e*x + d)*c + 2*c*d)*c*d - 693*((e*x + d)*c - 2*c*d)^6*sqrt(-(e*x + d)*c
+ 2*c*d))/(c^6*e^5))*c*e^5 - 9009*(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*sqrt(-(e*x + d)*c + 2*c*d))/c^2)*c*d^4)/e

Mupad [B] (verification not implemented)

Time = 10.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.58 \[ \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {16384\,c\,d^6\,\sqrt {c\,d^2-c\,e^2\,x^2}}{15015\,e\,\sqrt {d+e\,x}}-\frac {2\,c\,\sqrt {c\,d^2-c\,e^2\,x^2}\,\sqrt {d+e\,x}\,\left (1491\,d^5-4157\,d^4\,e\,x-5350\,d^3\,e^2\,x^2-70\,d^2\,e^3\,x^3+2835\,d\,e^4\,x^4+1155\,e^5\,x^5\right )}{15015\,e} \]

[In]

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

[Out]

- (16384*c*d^6*(c*d^2 - c*e^2*x^2)^(1/2))/(15015*e*(d + e*x)^(1/2)) - (2*c*(c*d^2 - c*e^2*x^2)^(1/2)*(d + e*x)
^(1/2)*(1491*d^5 + 1155*e^5*x^5 + 2835*d*e^4*x^4 - 5350*d^3*e^2*x^2 - 70*d^2*e^3*x^3 - 4157*d^4*e*x))/(15015*e
)