\(\int \frac {(a+c x^2)^3}{(d+e x)^{3/2}} \, dx\) [609]

Optimal result

Integrand size = 19, antiderivative size = 198 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2+a e^2\right )^3}{e^7 \sqrt {d+e x}}-\frac {12 c d \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{5/2}}{5 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {4 c^3 d (d+e x)^{9/2}}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \]

[Out]

2*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)*(e*x+d)^(3/2)/e^7-8/5*c^2*d*(3*a*e^2+5*c*d^2)*(e*x+d)^(5/2)/e^7+6/7*c^2*(a*e
^2+5*c*d^2)*(e*x+d)^(7/2)/e^7-4/3*c^3*d*(e*x+d)^(9/2)/e^7+2/11*c^3*(e*x+d)^(11/2)/e^7-2*(a*e^2+c*d^2)^3/e^7/(e
*x+d)^(1/2)-12*c*d*(a*e^2+c*d^2)^2*(e*x+d)^(1/2)/e^7

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {6 c^2 (d+e x)^{7/2} \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {8 c^2 d (d+e x)^{5/2} \left (3 a e^2+5 c d^2\right )}{5 e^7}+\frac {2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}-\frac {12 c d \sqrt {d+e x} \left (a e^2+c d^2\right )^2}{e^7}-\frac {2 \left (a e^2+c d^2\right )^3}{e^7 \sqrt {d+e x}}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7}-\frac {4 c^3 d (d+e x)^{9/2}}{3 e^7} \]

[In]

Int[(a + c*x^2)^3/(d + e*x)^(3/2),x]

[Out]

(-2*(c*d^2 + a*e^2)^3)/(e^7*Sqrt[d + e*x]) - (12*c*d*(c*d^2 + a*e^2)^2*Sqrt[d + e*x])/e^7 + (2*c*(c*d^2 + a*e^
2)*(5*c*d^2 + a*e^2)*(d + e*x)^(3/2))/e^7 - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(5/2))/(5*e^7) + (6*c^2*(5*
c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^7) - (4*c^3*d*(d + e*x)^(9/2))/(3*e^7) + (2*c^3*(d + e*x)^(11/2))/(11*e^7
)

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{3/2}}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 \sqrt {d+e x}}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \sqrt {d+e x}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{3/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{e^6}-\frac {6 c^3 d (d+e x)^{7/2}}{e^6}+\frac {c^3 (d+e x)^{9/2}}{e^6}\right ) \, dx \\ & = -\frac {2 \left (c d^2+a e^2\right )^3}{e^7 \sqrt {d+e x}}-\frac {12 c d \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^7}+\frac {2 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{5/2}}{5 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {4 c^3 d (d+e x)^{9/2}}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (1155 a^3 e^6+1155 a^2 c e^4 \left (8 d^2+4 d e x-e^2 x^2\right )+99 a c^2 e^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )+5 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )}{1155 e^7 \sqrt {d+e x}} \]

[In]

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

[Out]

(-2*(1155*a^3*e^6 + 1155*a^2*c*e^4*(8*d^2 + 4*d*e*x - e^2*x^2) + 99*a*c^2*e^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e
^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) + 5*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*
e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6)))/(1155*e^7*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.00 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(\frac {\left (210 e^{6} x^{6}-280 d \,e^{5} x^{5}+400 d^{2} e^{4} x^{4}-640 x^{3} d^{3} e^{3}+1280 d^{4} e^{2} x^{2}-5120 d^{5} e x -10240 d^{6}\right ) c^{3}-25344 \left (-\frac {5}{128} e^{4} x^{4}+\frac {1}{16} d \,e^{3} x^{3}-\frac {1}{8} d^{2} e^{2} x^{2}+\frac {1}{2} d^{3} e x +d^{4}\right ) e^{2} a \,c^{2}-18480 \left (-\frac {1}{8} x^{2} e^{2}+\frac {1}{2} d e x +d^{2}\right ) e^{4} a^{2} c -2310 e^{6} a^{3}}{1155 \sqrt {e x +d}\, e^{7}}\)	\(163\)
risch	\(-\frac {2 c \left (-105 c^{2} e^{5} x^{5}+245 c^{2} d \,e^{4} x^{4}-495 a c \,e^{5} x^{3}-445 d^{2} e^{3} x^{3} c^{2}+1287 a c d \,e^{4} x^{2}+765 c^{2} d^{3} e^{2} x^{2}-1155 a^{2} e^{5} x -2871 a c \,d^{2} e^{3} x -1405 c^{2} d^{4} e x +5775 a^{2} d \,e^{4}+9207 a c \,d^{3} e^{2}+3965 c^{2} d^{5}\right ) \sqrt {e x +d}}{1155 e^{7}}-\frac {2 \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{e^{7} \sqrt {e x +d}}\)	\(196\)
gosper	\(-\frac {2 \left (-105 x^{6} c^{3} e^{6}+140 x^{5} c^{3} d \,e^{5}-495 x^{4} a \,c^{2} e^{6}-200 x^{4} c^{3} d^{2} e^{4}+792 x^{3} a \,c^{2} d \,e^{5}+320 x^{3} c^{3} d^{3} e^{3}-1155 x^{2} a^{2} c \,e^{6}-1584 x^{2} a \,c^{2} d^{2} e^{4}-640 x^{2} c^{3} d^{4} e^{2}+4620 x \,a^{2} c d \,e^{5}+6336 x a \,c^{2} d^{3} e^{3}+2560 x \,c^{3} d^{5} e +1155 e^{6} a^{3}+9240 d^{2} e^{4} a^{2} c +12672 d^{4} e^{2} c^{2} a +5120 c^{3} d^{6}\right )}{1155 \sqrt {e x +d}\, e^{7}}\)	\(205\)
trager	\(-\frac {2 \left (-105 x^{6} c^{3} e^{6}+140 x^{5} c^{3} d \,e^{5}-495 x^{4} a \,c^{2} e^{6}-200 x^{4} c^{3} d^{2} e^{4}+792 x^{3} a \,c^{2} d \,e^{5}+320 x^{3} c^{3} d^{3} e^{3}-1155 x^{2} a^{2} c \,e^{6}-1584 x^{2} a \,c^{2} d^{2} e^{4}-640 x^{2} c^{3} d^{4} e^{2}+4620 x \,a^{2} c d \,e^{5}+6336 x a \,c^{2} d^{3} e^{3}+2560 x \,c^{3} d^{5} e +1155 e^{6} a^{3}+9240 d^{2} e^{4} a^{2} c +12672 d^{4} e^{2} c^{2} a +5120 c^{3} d^{6}\right )}{1155 \sqrt {e x +d}\, e^{7}}\)	\(205\)
derivativedivides	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {30 c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {24 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-8 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}+2 a^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+12 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}-12 a^{2} c d \,e^{4} \sqrt {e x +d}-24 a \,c^{2} d^{3} e^{2} \sqrt {e x +d}-12 c^{3} d^{5} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{\sqrt {e x +d}}}{e^{7}}\)	\(243\)
default	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {30 c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {24 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-8 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}+2 a^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+12 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+10 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}-12 a^{2} c d \,e^{4} \sqrt {e x +d}-24 a \,c^{2} d^{3} e^{2} \sqrt {e x +d}-12 c^{3} d^{5} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{\sqrt {e x +d}}}{e^{7}}\)	\(243\)

[In]

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

[Out]

1/1155*((210*e^6*x^6-280*d*e^5*x^5+400*d^2*e^4*x^4-640*d^3*e^3*x^3+1280*d^4*e^2*x^2-5120*d^5*e*x-10240*d^6)*c^
3-25344*(-5/128*e^4*x^4+1/16*d*e^3*x^3-1/8*d^2*e^2*x^2+1/2*d^3*e*x+d^4)*e^2*a*c^2-18480*(-1/8*x^2*e^2+1/2*d*e*
x+d^2)*e^4*a^2*c-2310*e^6*a^3)/(e*x+d)^(1/2)/e^7

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (105 \, c^{3} e^{6} x^{6} - 140 \, c^{3} d e^{5} x^{5} - 5120 \, c^{3} d^{6} - 12672 \, a c^{2} d^{4} e^{2} - 9240 \, a^{2} c d^{2} e^{4} - 1155 \, a^{3} e^{6} + 5 \, {\left (40 \, c^{3} d^{2} e^{4} + 99 \, a c^{2} e^{6}\right )} x^{4} - 8 \, {\left (40 \, c^{3} d^{3} e^{3} + 99 \, a c^{2} d e^{5}\right )} x^{3} + {\left (640 \, c^{3} d^{4} e^{2} + 1584 \, a c^{2} d^{2} e^{4} + 1155 \, a^{2} c e^{6}\right )} x^{2} - 4 \, {\left (640 \, c^{3} d^{5} e + 1584 \, a c^{2} d^{3} e^{3} + 1155 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{1155 \, {\left (e^{8} x + d e^{7}\right )}} \]

[In]

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

[Out]

2/1155*(105*c^3*e^6*x^6 - 140*c^3*d*e^5*x^5 - 5120*c^3*d^6 - 12672*a*c^2*d^4*e^2 - 9240*a^2*c*d^2*e^4 - 1155*a
^3*e^6 + 5*(40*c^3*d^2*e^4 + 99*a*c^2*e^6)*x^4 - 8*(40*c^3*d^3*e^3 + 99*a*c^2*d*e^5)*x^3 + (640*c^3*d^4*e^2 +
1584*a*c^2*d^2*e^4 + 1155*a^2*c*e^6)*x^2 - 4*(640*c^3*d^5*e + 1584*a*c^2*d^3*e^3 + 1155*a^2*c*d*e^5)*x)*sqrt(e
*x + d)/(e^8*x + d*e^7)

Sympy [A] (verification not implemented)

Time = 3.20 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {2 c^{3} d \left (d + e x\right )^{\frac {9}{2}}}{3 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{e^{6}} - \frac {\left (a e^{2} + c d^{2}\right )^{3}}{e^{6} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(-2*c**3*d*(d + e*x)**(9/2)/(3*e**6) + c**3*(d + e*x)**(11/2)/(11*e**6) + (d + e*x)**(7/2)*(3*a*c
**2*e**2 + 15*c**3*d**2)/(7*e**6) + (d + e*x)**(5/2)*(-12*a*c**2*d*e**2 - 20*c**3*d**3)/(5*e**6) + (d + e*x)**
(3/2)*(3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 + 15*c**3*d**4)/(3*e**6) + sqrt(d + e*x)*(-6*a**2*c*d*e**4 - 12*a*c
**2*d**3*e**2 - 6*c**3*d**5)/e**6 - (a*e**2 + c*d**2)**3/(e**6*sqrt(d + e*x)))/e, Ne(e, 0)), ((a**3*x + a**2*c
*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7)/d**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} - 770 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} d + 495 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 924 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 6930 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {1155 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}}{\sqrt {e x + d} e^{6}}\right )}}{1155 \, e} \]

[In]

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

[Out]

2/1155*((105*(e*x + d)^(11/2)*c^3 - 770*(e*x + d)^(9/2)*c^3*d + 495*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(7/2) -
924*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(5/2) + 1155*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*x + d)^(3/
2) - 6930*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*sqrt(e*x + d))/e^6 - 1155*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a
^2*c*d^2*e^4 + a^3*e^6)/(sqrt(e*x + d)*e^6))/e

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}}{\sqrt {e x + d} e^{7}} + \frac {2 \, {\left (105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} e^{70} - 770 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} d e^{70} + 2475 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} d^{2} e^{70} - 4620 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d^{3} e^{70} + 5775 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{4} e^{70} - 6930 \, \sqrt {e x + d} c^{3} d^{5} e^{70} + 495 \, {\left (e x + d\right )}^{\frac {7}{2}} a c^{2} e^{72} - 2772 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{2} d e^{72} + 6930 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} d^{2} e^{72} - 13860 \, \sqrt {e x + d} a c^{2} d^{3} e^{72} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c e^{74} - 6930 \, \sqrt {e x + d} a^{2} c d e^{74}\right )}}{1155 \, e^{77}} \]

[In]

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

[Out]

-2*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)/(sqrt(e*x + d)*e^7) + 2/1155*(105*(e*x + d)^(11/2)*
c^3*e^70 - 770*(e*x + d)^(9/2)*c^3*d*e^70 + 2475*(e*x + d)^(7/2)*c^3*d^2*e^70 - 4620*(e*x + d)^(5/2)*c^3*d^3*e
^70 + 5775*(e*x + d)^(3/2)*c^3*d^4*e^70 - 6930*sqrt(e*x + d)*c^3*d^5*e^70 + 495*(e*x + d)^(7/2)*a*c^2*e^72 - 2
772*(e*x + d)^(5/2)*a*c^2*d*e^72 + 6930*(e*x + d)^(3/2)*a*c^2*d^2*e^72 - 13860*sqrt(e*x + d)*a*c^2*d^3*e^72 +
1155*(e*x + d)^(3/2)*a^2*c*e^74 - 6930*sqrt(e*x + d)*a^2*c*d*e^74)/e^77

Mupad [B] (verification not implemented)

Time = 9.28 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{3\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}-\frac {2\,a^3\,e^6+6\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2+2\,c^3\,d^6}{e^7\,\sqrt {d+e\,x}}-\frac {4\,c^3\,d\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,\sqrt {d+e\,x}}{e^7} \]

[In]

int((a + c*x^2)^3/(d + e*x)^(3/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(7/2))/(7*e^7) + ((d + e*x)^(3/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2*d
^2*e^2))/(3*e^7) + (2*c^3*(d + e*x)^(11/2))/(11*e^7) - ((40*c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(5/2))/(5*e^7)
- (2*a^3*e^6 + 2*c^3*d^6 + 6*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4)/(e^7*(d + e*x)^(1/2)) - (4*c^3*d*(d + e*x)^(9/2
))/(3*e^7) - (12*c*d*(a*e^2 + c*d^2)^2*(d + e*x)^(1/2))/e^7