\(\int (d+e x)^{3/2} (c d^2-c e^2 x^2)^{3/2} \, dx\) [870]

Optimal result

Integrand size = 29, antiderivative size = 160 \[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e} \]

[Out]

-256/1155*d^3*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(5/2)-64/231*d^2*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(3/2)
-8/33*d*(-c*e^2*x^2+c*d^2)^(5/2)/c/e/(e*x+d)^(1/2)-2/11*(-c*e^2*x^2+c*d^2)^(5/2)*(e*x+d)^(1/2)/c/e

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}-\frac {256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}} \]

[In]

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

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(5/2))/(1155*c*e*(d + e*x)^(5/2)) - (64*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(231*c*e*
(d + e*x)^(3/2)) - (8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(33*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^
2)^(5/2))/(11*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 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac {1}{11} (12 d) \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx \\ & = -\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac {1}{33} \left (32 d^2\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx \\ & = -\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac {1}{231} \left (128 d^3\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx \\ & = -\frac {256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac {8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.46 \[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 c (d-e x)^2 \sqrt {c \left (d^2-e^2 x^2\right )} \left (533 d^3+755 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right )}{1155 e \sqrt {d+e x}} \]

[In]

Integrate[(d + e*x)^(3/2)*(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)]*(533*d^3 + 755*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3))/(1155*e*Sqrt[
d + e*x])

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.41

method	result	size
gosper	\(-\frac {2 \left (-e x +d \right ) \left (105 e^{3} x^{3}+455 d \,e^{2} x^{2}+755 d^{2} e x +533 d^{3}\right ) \left (-c \,x^{2} e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{1155 e \left (e x +d \right )^{\frac {3}{2}}}\)	\(66\)
default	\(-\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, c \left (-e x +d \right )^{2} \left (105 e^{3} x^{3}+455 d \,e^{2} x^{2}+755 d^{2} e x +533 d^{3}\right )}{1155 \sqrt {e x +d}\, e}\)	\(68\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2} \left (105 e^{5} x^{5}+245 x^{4} d \,e^{4}-50 d^{2} e^{3} x^{3}-522 d^{3} e^{2} x^{2}-311 d^{4} e x +533 d^{5}\right ) \left (-e x +d \right )}{1155 \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(129\)

[In]

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

[Out]

-2/1155*(-e*x+d)*(105*e^3*x^3+455*d*e^2*x^2+755*d^2*e*x+533*d^3)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.59 \[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (105 \, c e^{5} x^{5} + 245 \, c d e^{4} x^{4} - 50 \, c d^{2} e^{3} x^{3} - 522 \, c d^{3} e^{2} x^{2} - 311 \, c d^{4} e x + 533 \, c d^{5}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{1155 \, {\left (e^{2} x + d e\right )}} \]

[In]

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

[Out]

-2/1155*(105*c*e^5*x^5 + 245*c*d*e^4*x^4 - 50*c*d^2*e^3*x^3 - 522*c*d^3*e^2*x^2 - 311*c*d^4*e*x + 533*c*d^5)*s
qrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e)

Sympy [F]

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

[In]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.60 \[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (105 \, c^{\frac {3}{2}} e^{5} x^{5} + 245 \, c^{\frac {3}{2}} d e^{4} x^{4} - 50 \, c^{\frac {3}{2}} d^{2} e^{3} x^{3} - 522 \, c^{\frac {3}{2}} d^{3} e^{2} x^{2} - 311 \, c^{\frac {3}{2}} d^{4} e x + 533 \, c^{\frac {3}{2}} d^{5}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{1155 \, {\left (e^{2} x + d e\right )}} \]

[In]

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

[Out]

-2/1155*(105*c^(3/2)*e^5*x^5 + 245*c^(3/2)*d*e^4*x^4 - 50*c^(3/2)*d^2*e^3*x^3 - 522*c^(3/2)*d^3*e^2*x^2 - 311*
c^(3/2)*d^4*e*x + 533*c^(3/2)*d^5)*(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. 468 vs. \(2 (136) = 272\).

Time = 0.30 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.92 \[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, {\left (2 \, \sqrt {2} \sqrt {c d} d - \frac {{\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c}\right )} c d^{4} + 22 \, {\left (\frac {26 \, \sqrt {2} \sqrt {c d} d^{4}}{e^{3}} + \frac {105 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{3} d^{3} - 189 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{2} d^{2} - 135 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - 35 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{4} e^{3}}\right )} c d e^{3} - {\left (\frac {422 \, \sqrt {2} \sqrt {c d} d^{5}}{e^{4}} - \frac {1155 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{4} d^{4} - 2772 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{3} d^{3} - 2970 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{2} d^{2} - 1540 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{4} \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c d - 315 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{5} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{5} e^{4}}\right )} c e^{4} - 462 \, {\left (2 \, \sqrt {2} \sqrt {c d} d^{2} + \frac {5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c^{2}}\right )} c d^{3}\right )}}{3465 \, e} \]

[In]

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

[Out]

2/3465*(1155*(2*sqrt(2)*sqrt(c*d)*d - (-(e*x + d)*c + 2*c*d)^(3/2)/c)*c*d^4 + 22*(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*e^3 - (422*sqrt(2)*sqrt(c*d)*d^5/e^4 - (1155*(-(e*x + d)*c + 2*c*d)^(3/2)*c^4*d^4 - 277
2*((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*s
qrt(-(e*x + d)*c + 2*c*d))/(c^5*e^4))*c*e^4 - 462*(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^3)/e

Mupad [B] (verification not implemented)

Time = 10.53 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.80 \[ \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {1066\,c\,d^5\,\sqrt {d+e\,x}}{1155\,e^2}-\frac {348\,c\,d^3\,x^2\,\sqrt {d+e\,x}}{385}+\frac {2\,c\,e^3\,x^5\,\sqrt {d+e\,x}}{11}-\frac {20\,c\,d^2\,e\,x^3\,\sqrt {d+e\,x}}{231}-\frac {622\,c\,d^4\,x\,\sqrt {d+e\,x}}{1155\,e}+\frac {14\,c\,d\,e^2\,x^4\,\sqrt {d+e\,x}}{33}\right )}{x+\frac {d}{e}} \]

[In]

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

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((1066*c*d^5*(d + e*x)^(1/2))/(1155*e^2) - (348*c*d^3*x^2*(d + e*x)^(1/2))/385 + (
2*c*e^3*x^5*(d + e*x)^(1/2))/11 - (20*c*d^2*e*x^3*(d + e*x)^(1/2))/231 - (622*c*d^4*x*(d + e*x)^(1/2))/(1155*e
) + (14*c*d*e^2*x^4*(d + e*x)^(1/2))/33))/(x + d/e)