\(\int \sqrt {d+e x} (a+c x^2)^3 \, dx\) [607]

Optimal result

Integrand size = 19, antiderivative size = 204 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \]

[Out]

2/3*(a*e^2+c*d^2)^3*(e*x+d)^(3/2)/e^7-12/5*c*d*(a*e^2+c*d^2)^2*(e*x+d)^(5/2)/e^7+6/7*c*(a*e^2+c*d^2)*(a*e^2+5*
c*d^2)*(e*x+d)^(7/2)/e^7-8/9*c^2*d*(3*a*e^2+5*c*d^2)*(e*x+d)^(9/2)/e^7+6/11*c^2*(a*e^2+5*c*d^2)*(e*x+d)^(11/2)
/e^7-12/13*c^3*d*(e*x+d)^(13/2)/e^7+2/15*c^3*(e*x+d)^(15/2)/e^7

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 204, 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} \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {6 c^2 (d+e x)^{11/2} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {8 c^2 d (d+e x)^{9/2} \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {6 c (d+e x)^{7/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {12 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^3}{3 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7} \]

[In]

Int[Sqrt[d + e*x]*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(3/2))/(3*e^7) - (12*c*d*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^7) + (6*c*(c*d
^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(9/2))/(9*e^7)
+ (6*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^7) - (12*c^3*d*(d + e*x)^(13/2))/(13*e^7) + (2*c^3*(d + e*
x)^(15/2))/(15*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 \sqrt {d+e x}}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{7/2}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^6}-\frac {6 c^3 d (d+e x)^{11/2}}{e^6}+\frac {c^3 (d+e x)^{13/2}}{e^6}\right ) \, dx \\ & = \frac {2 \left (c d^2+a e^2\right )^3 (d+e x)^{3/2}}{3 e^7}-\frac {12 c d \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^7}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{9/2}}{9 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac {12 c^3 d (d+e x)^{13/2}}{13 e^7}+\frac {2 c^3 (d+e x)^{15/2}}{15 e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 (d+e x)^{3/2} \left (15015 a^3 e^6+1287 a^2 c e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+39 a c^2 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+c^3 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7} \]

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(15015*a^3*e^6 + 1287*a^2*c*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 39*a*c^2*e^2*(128*d^4 - 1
92*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + c^3*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2
- 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7)

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {2 \left (\left (\frac {1}{5} e^{6} x^{6}-\frac {12}{65} d \,e^{5} x^{5}+\frac {24}{143} d^{2} e^{4} x^{4}-\frac {64}{429} x^{3} d^{3} e^{3}+\frac {128}{1001} d^{4} e^{2} x^{2}-\frac {512}{5005} d^{5} e x +\frac {1024}{15015} d^{6}\right ) c^{3}+\frac {128 \left (\frac {315}{128} e^{4} x^{4}-\frac {35}{16} d \,e^{3} x^{3}+\frac {15}{8} d^{2} e^{2} x^{2}-\frac {3}{2} d^{3} e x +d^{4}\right ) e^{2} a \,c^{2}}{385}+\frac {24 \left (\frac {15}{8} x^{2} e^{2}-\frac {3}{2} d e x +d^{2}\right ) e^{4} a^{2} c}{35}+e^{6} a^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{7}}\)	\(162\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 x^{6} c^{3} e^{6}-2772 x^{5} c^{3} d \,e^{5}+12285 x^{4} a \,c^{2} e^{6}+2520 x^{4} c^{3} d^{2} e^{4}-10920 x^{3} a \,c^{2} d \,e^{5}-2240 x^{3} c^{3} d^{3} e^{3}+19305 x^{2} a^{2} c \,e^{6}+9360 x^{2} a \,c^{2} d^{2} e^{4}+1920 x^{2} c^{3} d^{4} e^{2}-15444 x \,a^{2} c d \,e^{5}-7488 x a \,c^{2} d^{3} e^{3}-1536 x \,c^{3} d^{5} e +15015 e^{6} a^{3}+10296 d^{2} e^{4} a^{2} c +4992 d^{4} e^{2} c^{2} a +1024 c^{3} d^{6}\right )}{45045 e^{7}}\)	\(205\)
trager	\(\frac {2 \left (3003 c^{3} e^{7} x^{7}+231 c^{3} d \,e^{6} x^{6}+12285 e^{7} c^{2} a \,x^{5}-252 c^{3} d^{2} e^{5} x^{5}+1365 d \,e^{6} c^{2} a \,x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+3861 a^{2} c d \,e^{6} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x -5148 a^{2} c \,d^{2} e^{5} x -2496 a \,c^{2} d^{4} e^{3} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}+10296 a^{2} c \,d^{3} e^{4}+4992 a \,c^{2} d^{5} e^{2}+1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\)	\(259\)
risch	\(\frac {2 \left (3003 c^{3} e^{7} x^{7}+231 c^{3} d \,e^{6} x^{6}+12285 e^{7} c^{2} a \,x^{5}-252 c^{3} d^{2} e^{5} x^{5}+1365 d \,e^{6} c^{2} a \,x^{4}+280 c^{3} d^{3} e^{4} x^{4}+19305 a^{2} c \,e^{7} x^{3}-1560 a \,c^{2} d^{2} e^{5} x^{3}-320 c^{3} d^{4} e^{3} x^{3}+3861 a^{2} c d \,e^{6} x^{2}+1872 a \,c^{2} d^{3} e^{4} x^{2}+384 c^{3} d^{5} e^{2} x^{2}+15015 a^{3} e^{7} x -5148 a^{2} c \,d^{2} e^{5} x -2496 a \,c^{2} d^{4} e^{3} x -512 c^{3} d^{6} e x +15015 a^{3} d \,e^{6}+10296 a^{2} c \,d^{3} e^{4}+4992 a \,c^{2} d^{5} e^{2}+1024 c^{3} d^{7}\right ) \sqrt {e x +d}}{45045 e^{7}}\)	\(259\)
derivativedivides	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\)	\(269\)
default	\(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-8 \left (e^{2} a +c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (e^{2} a +c \,d^{2}\right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )+c \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 \left (e^{2} a +c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{7}}\)	\(269\)

[In]

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

[Out]

2/3*((1/5*e^6*x^6-12/65*d*e^5*x^5+24/143*d^2*e^4*x^4-64/429*x^3*d^3*e^3+128/1001*d^4*e^2*x^2-512/5005*d^5*e*x+
1024/15015*d^6)*c^3+128/385*(315/128*e^4*x^4-35/16*d*e^3*x^3+15/8*d^2*e^2*x^2-3/2*d^3*e*x+d^4)*e^2*a*c^2+24/35
*(15/8*x^2*e^2-3/2*d*e*x+d^2)*e^4*a^2*c+e^6*a^3)*(e*x+d)^(3/2)/e^7

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.24 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, c^{3} e^{7} x^{7} + 231 \, c^{3} d e^{6} x^{6} + 1024 \, c^{3} d^{7} + 4992 \, a c^{2} d^{5} e^{2} + 10296 \, a^{2} c d^{3} e^{4} + 15015 \, a^{3} d e^{6} - 63 \, {\left (4 \, c^{3} d^{2} e^{5} - 195 \, a c^{2} e^{7}\right )} x^{5} + 35 \, {\left (8 \, c^{3} d^{3} e^{4} + 39 \, a c^{2} d e^{6}\right )} x^{4} - 5 \, {\left (64 \, c^{3} d^{4} e^{3} + 312 \, a c^{2} d^{2} e^{5} - 3861 \, a^{2} c e^{7}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{5} e^{2} + 624 \, a c^{2} d^{3} e^{4} + 1287 \, a^{2} c d e^{6}\right )} x^{2} - {\left (512 \, c^{3} d^{6} e + 2496 \, a c^{2} d^{4} e^{3} + 5148 \, a^{2} c d^{2} e^{5} - 15015 \, a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \]

[In]

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

[Out]

2/45045*(3003*c^3*e^7*x^7 + 231*c^3*d*e^6*x^6 + 1024*c^3*d^7 + 4992*a*c^2*d^5*e^2 + 10296*a^2*c*d^3*e^4 + 1501
5*a^3*d*e^6 - 63*(4*c^3*d^2*e^5 - 195*a*c^2*e^7)*x^5 + 35*(8*c^3*d^3*e^4 + 39*a*c^2*d*e^6)*x^4 - 5*(64*c^3*d^4
*e^3 + 312*a*c^2*d^2*e^5 - 3861*a^2*c*e^7)*x^3 + 3*(128*c^3*d^5*e^2 + 624*a*c^2*d^3*e^4 + 1287*a^2*c*d*e^6)*x^
2 - (512*c^3*d^6*e + 2496*a*c^2*d^4*e^3 + 5148*a^2*c*d^2*e^5 - 15015*a^3*e^7)*x)*sqrt(e*x + d)/e^7

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.49 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (- \frac {6 c^{3} d \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 6 a^{2} c d e^{4} - 12 a c^{2} d^{3} e^{2} - 6 c^{3} d^{5}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}{3 e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(-6*c**3*d*(d + e*x)**(13/2)/(13*e**6) + c**3*(d + e*x)**(15/2)/(15*e**6) + (d + e*x)**(11/2)*(3*
a*c**2*e**2 + 15*c**3*d**2)/(11*e**6) + (d + e*x)**(9/2)*(-12*a*c**2*d*e**2 - 20*c**3*d**3)/(9*e**6) + (d + e*
x)**(7/2)*(3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 + 15*c**3*d**4)/(7*e**6) + (d + e*x)**(5/2)*(-6*a**2*c*d*e**4 -
12*a*c**2*d**3*e**2 - 6*c**3*d**5)/(5*e**6) + (d + e*x)**(3/2)*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**
4*e**2 + c**3*d**6)/(3*e**6))/e, Ne(e, 0)), (sqrt(d)*(a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7), T
rue))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.02 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 20790 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{3} d + 12285 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 20020 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 19305 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 54054 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{7}} \]

[In]

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^3 - 20790*(e*x + d)^(13/2)*c^3*d + 12285*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(1
1/2) - 20020*(5*c^3*d^3 + 3*a*c^2*d*e^2)*(e*x + d)^(9/2) + 19305*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*
x + d)^(7/2) - 54054*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(5/2) + 15015*(c^3*d^6 + 3*a*c^2*d^4*
e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e*x + d)^(3/2))/e^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (176) = 352\).

Time = 0.27 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.30 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {2 \, {\left (45045 \, \sqrt {e x + d} a^{3} d + 15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{3} + \frac {9009 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} c d}{e^{2}} + \frac {3861 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} c}{e^{2}} + \frac {429 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a c^{2} d}{e^{4}} + \frac {195 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a c^{2}}{e^{4}} + \frac {15 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3} d}{e^{6}} + \frac {7 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} - 3465 \, {\left (e x + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (e x + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {e x + d} d^{7}\right )} c^{3}}{e^{6}}\right )}}{45045 \, e} \]

[In]

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

[Out]

2/45045*(45045*sqrt(e*x + d)*a^3*d + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3 + 9009*(3*(e*x + d)^(5/2)
- 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*c*d/e^2 + 3861*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d +
35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*c/e^2 + 429*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d +
378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*c^2*d/e^4 + 195*(63*(e*x + d)^(11
/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 -
693*sqrt(e*x + d)*d^5)*a*c^2/e^4 + 15*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d
^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*
c^3*d/e^6 + 7*(429*(e*x + d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(
9/2)*d^3 + 32175*(e*x + d)^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x +
d)*d^7)*c^3/e^6)/e

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.92 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^3 \, dx=\frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{7\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^7}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {12\,c\,d\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \]

[In]

int((a + c*x^2)^3*(d + e*x)^(1/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(11/2))/(11*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^4 + 6*a^2*c*e^4 + 36*a*c^2
*d^2*e^2))/(7*e^7) + (2*c^3*(d + e*x)^(15/2))/(15*e^7) + (2*(a*e^2 + c*d^2)^3*(d + e*x)^(3/2))/(3*e^7) - ((40*
c^3*d^3 + 24*a*c^2*d*e^2)*(d + e*x)^(9/2))/(9*e^7) - (12*c^3*d*(d + e*x)^(13/2))/(13*e^7) - (12*c*d*(a*e^2 + c
*d^2)^2*(d + e*x)^(5/2))/(5*e^7)