3.534 \(\int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=307 \[ \frac{x \left (a+c x^2\right )^{5/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{480 c^2}+\frac{a x \left (a+c x^2\right )^{3/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{384 c^2}+\frac{a^2 x \sqrt{a+c x^2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^2}+\frac{a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}+\frac{e \left (a+c x^2\right )^{7/2} \left (7 e x \left (116 c d^2-27 a e^2\right )+16 d \left (103 c d^2-40 a e^2\right )\right )}{5040 c^2}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^3}{10 c}+\frac{13 d e \left (a+c x^2\right )^{7/2} (d+e x)^2}{90 c} \]

[Out]

(a^2*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*Sqrt[a + c*x^2])/(256*c^2) + (a
*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*(a + c*x^2)^(3/2))/(384*c^2) + ((80
*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*(a + c*x^2)^(5/2))/(480*c^2) + (13*d*e*
(d + e*x)^2*(a + c*x^2)^(7/2))/(90*c) + (e*(d + e*x)^3*(a + c*x^2)^(7/2))/(10*c)
+ (e*(16*d*(103*c*d^2 - 40*a*e^2) + 7*e*(116*c*d^2 - 27*a*e^2)*x)*(a + c*x^2)^(
7/2))/(5040*c^2) + (a^3*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[
c]*x)/Sqrt[a + c*x^2]])/(256*c^(5/2))

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Rubi [A]  time = 0.749871, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{x \left (a+c x^2\right )^{5/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{480 c^2}+\frac{a x \left (a+c x^2\right )^{3/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{384 c^2}+\frac{a^2 x \sqrt{a+c x^2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^2}+\frac{a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}+\frac{e \left (a+c x^2\right )^{7/2} \left (7 e x \left (116 c d^2-27 a e^2\right )+16 d \left (103 c d^2-40 a e^2\right )\right )}{5040 c^2}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^3}{10 c}+\frac{13 d e \left (a+c x^2\right )^{7/2} (d+e x)^2}{90 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4*(a + c*x^2)^(5/2),x]

[Out]

(a^2*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*Sqrt[a + c*x^2])/(256*c^2) + (a
*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*(a + c*x^2)^(3/2))/(384*c^2) + ((80
*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*(a + c*x^2)^(5/2))/(480*c^2) + (13*d*e*
(d + e*x)^2*(a + c*x^2)^(7/2))/(90*c) + (e*(d + e*x)^3*(a + c*x^2)^(7/2))/(10*c)
+ (e*(16*d*(103*c*d^2 - 40*a*e^2) + 7*e*(116*c*d^2 - 27*a*e^2)*x)*(a + c*x^2)^(
7/2))/(5040*c^2) + (a^3*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[
c]*x)/Sqrt[a + c*x^2]])/(256*c^(5/2))

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Rubi in Sympy [A]  time = 63.138, size = 298, normalized size = 0.97 \[ \frac{a^{3} \left (3 a^{2} e^{4} - 60 a c d^{2} e^{2} + 80 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{256 c^{\frac{5}{2}}} + \frac{a^{2} x \sqrt{a + c x^{2}} \left (3 a^{2} e^{4} - 60 a c d^{2} e^{2} + 80 c^{2} d^{4}\right )}{256 c^{2}} + \frac{a x \left (a + c x^{2}\right )^{\frac{3}{2}} \left (3 a^{2} e^{4} - 60 a c d^{2} e^{2} + 80 c^{2} d^{4}\right )}{384 c^{2}} + \frac{13 d e \left (a + c x^{2}\right )^{\frac{7}{2}} \left (d + e x\right )^{2}}{90 c} + \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}} \left (d + e x\right )^{3}}{10 c} - \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}} \left (d \left (640 a e^{2} - 1648 c d^{2}\right ) + 7 e x \left (27 a e^{2} - 116 c d^{2}\right )\right )}{5040 c^{2}} + \frac{x \left (a + c x^{2}\right )^{\frac{5}{2}} \left (3 a^{2} e^{4} - 60 a c d^{2} e^{2} + 80 c^{2} d^{4}\right )}{480 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(c*x**2+a)**(5/2),x)

[Out]

a**3*(3*a**2*e**4 - 60*a*c*d**2*e**2 + 80*c**2*d**4)*atanh(sqrt(c)*x/sqrt(a + c*
x**2))/(256*c**(5/2)) + a**2*x*sqrt(a + c*x**2)*(3*a**2*e**4 - 60*a*c*d**2*e**2
+ 80*c**2*d**4)/(256*c**2) + a*x*(a + c*x**2)**(3/2)*(3*a**2*e**4 - 60*a*c*d**2*
e**2 + 80*c**2*d**4)/(384*c**2) + 13*d*e*(a + c*x**2)**(7/2)*(d + e*x)**2/(90*c)
+ e*(a + c*x**2)**(7/2)*(d + e*x)**3/(10*c) - e*(a + c*x**2)**(7/2)*(d*(640*a*e
**2 - 1648*c*d**2) + 7*e*x*(27*a*e**2 - 116*c*d**2))/(5040*c**2) + x*(a + c*x**2
)**(5/2)*(3*a**2*e**4 - 60*a*c*d**2*e**2 + 80*c**2*d**4)/(480*c**2)

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Mathematica [A]  time = 0.459993, size = 283, normalized size = 0.92 \[ \frac{a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{256 c^{5/2}}+\frac{\sqrt{a+c x^2} \left (-5 a^4 e^3 (2048 d+189 e x)+10 a^3 c e \left (4608 d^3+1890 d^2 e x+512 d e^2 x^2+63 e^3 x^3\right )+24 a^2 c^2 x \left (2310 d^4+5760 d^3 e x+6195 d^2 e^2 x^2+3200 d e^3 x^3+651 e^4 x^4\right )+16 a c^3 x^3 \left (2730 d^4+8640 d^3 e x+10710 d^2 e^2 x^2+6080 d e^3 x^3+1323 e^4 x^4\right )+64 c^4 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )}{80640 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[a + c*x^2]*(-5*a^4*e^3*(2048*d + 189*e*x) + 10*a^3*c*e*(4608*d^3 + 1890*d^
2*e*x + 512*d*e^2*x^2 + 63*e^3*x^3) + 64*c^4*x^5*(210*d^4 + 720*d^3*e*x + 945*d^
2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + 24*a^2*c^2*x*(2310*d^4 + 5760*d^3*e*x
+ 6195*d^2*e^2*x^2 + 3200*d*e^3*x^3 + 651*e^4*x^4) + 16*a*c^3*x^3*(2730*d^4 + 8
640*d^3*e*x + 10710*d^2*e^2*x^2 + 6080*d*e^3*x^3 + 1323*e^4*x^4)))/(80640*c^2) +
(a^3*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2
]])/(256*c^(5/2))

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Maple [A]  time = 0.019, size = 386, normalized size = 1.3 \[{\frac{{d}^{4}x}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{4}ax}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{4}{a}^{2}x}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{d}^{4}{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{4}{x}^{3}}{10\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{e}^{4}ax}{80\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}{e}^{4}x}{160\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{4}{a}^{3}x}{128\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{4}{a}^{4}x}{256\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,{e}^{4}{a}^{5}}{256}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{4\,d{e}^{3}{x}^{2}}{9\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,d{e}^{3}a}{63\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{d}^{2}{e}^{2}x}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{d}^{2}{e}^{2}ax}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{2}{e}^{2}{a}^{2}x}{32\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{d}^{2}{e}^{2}{a}^{3}x}{64\,c}\sqrt{c{x}^{2}+a}}-{\frac{15\,{d}^{2}{e}^{2}{a}^{4}}{64}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{4\,{d}^{3}e}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(c*x^2+a)^(5/2),x)

[Out]

1/6*d^4*x*(c*x^2+a)^(5/2)+5/24*d^4*a*x*(c*x^2+a)^(3/2)+5/16*d^4*a^2*x*(c*x^2+a)^
(1/2)+5/16*d^4*a^3/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/10*e^4*x^3*(c*x^2+a)^
(7/2)/c-3/80*e^4*a/c^2*x*(c*x^2+a)^(7/2)+1/160*e^4*a^2/c^2*x*(c*x^2+a)^(5/2)+1/1
28*e^4*a^3/c^2*x*(c*x^2+a)^(3/2)+3/256*e^4*a^4/c^2*x*(c*x^2+a)^(1/2)+3/256*e^4*a
^5/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+4/9*d*e^3*x^2*(c*x^2+a)^(7/2)/c-8/63*d*
e^3*a/c^2*(c*x^2+a)^(7/2)+3/4*d^2*e^2*x*(c*x^2+a)^(7/2)/c-1/8*d^2*e^2*a/c*x*(c*x
^2+a)^(5/2)-5/32*d^2*e^2*a^2/c*x*(c*x^2+a)^(3/2)-15/64*d^2*e^2*a^3/c*x*(c*x^2+a)
^(1/2)-15/64*d^2*e^2*a^4/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+4/7*d^3*e*(c*x^2+
a)^(7/2)/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275818, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (8064 \, c^{4} e^{4} x^{9} + 35840 \, c^{4} d e^{3} x^{8} + 46080 \, a^{3} c d^{3} e - 10240 \, a^{4} d e^{3} + 3024 \,{\left (20 \, c^{4} d^{2} e^{2} + 7 \, a c^{3} e^{4}\right )} x^{7} + 5120 \,{\left (9 \, c^{4} d^{3} e + 19 \, a c^{3} d e^{3}\right )} x^{6} + 168 \,{\left (80 \, c^{4} d^{4} + 1020 \, a c^{3} d^{2} e^{2} + 93 \, a^{2} c^{2} e^{4}\right )} x^{5} + 15360 \,{\left (9 \, a c^{3} d^{3} e + 5 \, a^{2} c^{2} d e^{3}\right )} x^{4} + 210 \,{\left (208 \, a c^{3} d^{4} + 708 \, a^{2} c^{2} d^{2} e^{2} + 3 \, a^{3} c e^{4}\right )} x^{3} + 5120 \,{\left (27 \, a^{2} c^{2} d^{3} e + a^{3} c d e^{3}\right )} x^{2} + 315 \,{\left (176 \, a^{2} c^{2} d^{4} + 60 \, a^{3} c d^{2} e^{2} - 3 \, a^{4} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 315 \,{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{161280 \, c^{\frac{5}{2}}}, \frac{{\left (8064 \, c^{4} e^{4} x^{9} + 35840 \, c^{4} d e^{3} x^{8} + 46080 \, a^{3} c d^{3} e - 10240 \, a^{4} d e^{3} + 3024 \,{\left (20 \, c^{4} d^{2} e^{2} + 7 \, a c^{3} e^{4}\right )} x^{7} + 5120 \,{\left (9 \, c^{4} d^{3} e + 19 \, a c^{3} d e^{3}\right )} x^{6} + 168 \,{\left (80 \, c^{4} d^{4} + 1020 \, a c^{3} d^{2} e^{2} + 93 \, a^{2} c^{2} e^{4}\right )} x^{5} + 15360 \,{\left (9 \, a c^{3} d^{3} e + 5 \, a^{2} c^{2} d e^{3}\right )} x^{4} + 210 \,{\left (208 \, a c^{3} d^{4} + 708 \, a^{2} c^{2} d^{2} e^{2} + 3 \, a^{3} c e^{4}\right )} x^{3} + 5120 \,{\left (27 \, a^{2} c^{2} d^{3} e + a^{3} c d e^{3}\right )} x^{2} + 315 \,{\left (176 \, a^{2} c^{2} d^{4} + 60 \, a^{3} c d^{2} e^{2} - 3 \, a^{4} e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 315 \,{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{80640 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d)^4,x, algorithm="fricas")

[Out]

[1/161280*(2*(8064*c^4*e^4*x^9 + 35840*c^4*d*e^3*x^8 + 46080*a^3*c*d^3*e - 10240
*a^4*d*e^3 + 3024*(20*c^4*d^2*e^2 + 7*a*c^3*e^4)*x^7 + 5120*(9*c^4*d^3*e + 19*a*
c^3*d*e^3)*x^6 + 168*(80*c^4*d^4 + 1020*a*c^3*d^2*e^2 + 93*a^2*c^2*e^4)*x^5 + 15
360*(9*a*c^3*d^3*e + 5*a^2*c^2*d*e^3)*x^4 + 210*(208*a*c^3*d^4 + 708*a^2*c^2*d^2
*e^2 + 3*a^3*c*e^4)*x^3 + 5120*(27*a^2*c^2*d^3*e + a^3*c*d*e^3)*x^2 + 315*(176*a
^2*c^2*d^4 + 60*a^3*c*d^2*e^2 - 3*a^4*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) + 315*(80*
a^3*c^2*d^4 - 60*a^4*c*d^2*e^2 + 3*a^5*e^4)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^
2 + a)*sqrt(c)))/c^(5/2), 1/80640*((8064*c^4*e^4*x^9 + 35840*c^4*d*e^3*x^8 + 460
80*a^3*c*d^3*e - 10240*a^4*d*e^3 + 3024*(20*c^4*d^2*e^2 + 7*a*c^3*e^4)*x^7 + 512
0*(9*c^4*d^3*e + 19*a*c^3*d*e^3)*x^6 + 168*(80*c^4*d^4 + 1020*a*c^3*d^2*e^2 + 93
*a^2*c^2*e^4)*x^5 + 15360*(9*a*c^3*d^3*e + 5*a^2*c^2*d*e^3)*x^4 + 210*(208*a*c^3
*d^4 + 708*a^2*c^2*d^2*e^2 + 3*a^3*c*e^4)*x^3 + 5120*(27*a^2*c^2*d^3*e + a^3*c*d
*e^3)*x^2 + 315*(176*a^2*c^2*d^4 + 60*a^3*c*d^2*e^2 - 3*a^4*e^4)*x)*sqrt(c*x^2 +
a)*sqrt(-c) + 315*(80*a^3*c^2*d^4 - 60*a^4*c*d^2*e^2 + 3*a^5*e^4)*arctan(sqrt(-
c)*x/sqrt(c*x^2 + a)))/(sqrt(-c)*c^2)]

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Sympy [A]  time = 177.261, size = 1062, normalized size = 3.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4*(c*x**2+a)**(5/2),x)

[Out]

-3*a**(9/2)*e**4*x/(256*c**2*sqrt(1 + c*x**2/a)) + 15*a**(7/2)*d**2*e**2*x/(64*c
*sqrt(1 + c*x**2/a)) - a**(7/2)*e**4*x**3/(256*c*sqrt(1 + c*x**2/a)) + a**(5/2)*
d**4*x*sqrt(1 + c*x**2/a)/2 + 3*a**(5/2)*d**4*x/(16*sqrt(1 + c*x**2/a)) + 133*a*
*(5/2)*d**2*e**2*x**3/(64*sqrt(1 + c*x**2/a)) + 129*a**(5/2)*e**4*x**5/(640*sqrt
(1 + c*x**2/a)) + 35*a**(3/2)*c*d**4*x**3/(48*sqrt(1 + c*x**2/a)) + 127*a**(3/2)
*c*d**2*e**2*x**5/(32*sqrt(1 + c*x**2/a)) + 73*a**(3/2)*c*e**4*x**7/(160*sqrt(1
+ c*x**2/a)) + 17*sqrt(a)*c**2*d**4*x**5/(24*sqrt(1 + c*x**2/a)) + 23*sqrt(a)*c*
*2*d**2*e**2*x**7/(8*sqrt(1 + c*x**2/a)) + 29*sqrt(a)*c**2*e**4*x**9/(80*sqrt(1
+ c*x**2/a)) + 3*a**5*e**4*asinh(sqrt(c)*x/sqrt(a))/(256*c**(5/2)) - 15*a**4*d**
2*e**2*asinh(sqrt(c)*x/sqrt(a))/(64*c**(3/2)) + 5*a**3*d**4*asinh(sqrt(c)*x/sqrt
(a))/(16*sqrt(c)) + 4*a**2*d**3*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*
x**2)**(3/2)/(3*c), True)) + 4*a**2*d*e**3*Piecewise((-2*a**2*sqrt(a + c*x**2)/(
15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)),
(sqrt(a)*x**4/4, True)) + 8*a*c*d**3*e*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c
**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqr
t(a)*x**4/4, True)) + 8*a*c*d*e**3*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3)
- 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x*
*6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True)) + 4*c**2*d**3*e*Piecew
ise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2
) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a
)*x**6/6, True)) + 4*c**2*d*e**3*Piecewise((-16*a**4*sqrt(a + c*x**2)/(315*c**4)
+ 8*a**3*x**2*sqrt(a + c*x**2)/(315*c**3) - 2*a**2*x**4*sqrt(a + c*x**2)/(105*c
**2) + a*x**6*sqrt(a + c*x**2)/(63*c) + x**8*sqrt(a + c*x**2)/9, Ne(c, 0)), (sqr
t(a)*x**8/8, True)) + c**3*d**4*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + 3*c**3*d**
2*e**2*x**9/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**3*e**4*x**11/(10*sqrt(a)*sqrt(1
+ c*x**2/a))

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GIAC/XCAS [A]  time = 0.220076, size = 486, normalized size = 1.58 \[ \frac{1}{80640} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (2 \,{\left (7 \,{\left (8 \,{\left (9 \, c^{2} x e^{4} + 40 \, c^{2} d e^{3}\right )} x + \frac{27 \,{\left (20 \, c^{10} d^{2} e^{2} + 7 \, a c^{9} e^{4}\right )}}{c^{8}}\right )} x + \frac{320 \,{\left (9 \, c^{10} d^{3} e + 19 \, a c^{9} d e^{3}\right )}}{c^{8}}\right )} x + \frac{21 \,{\left (80 \, c^{10} d^{4} + 1020 \, a c^{9} d^{2} e^{2} + 93 \, a^{2} c^{8} e^{4}\right )}}{c^{8}}\right )} x + \frac{1920 \,{\left (9 \, a c^{9} d^{3} e + 5 \, a^{2} c^{8} d e^{3}\right )}}{c^{8}}\right )} x + \frac{105 \,{\left (208 \, a c^{9} d^{4} + 708 \, a^{2} c^{8} d^{2} e^{2} + 3 \, a^{3} c^{7} e^{4}\right )}}{c^{8}}\right )} x + \frac{2560 \,{\left (27 \, a^{2} c^{8} d^{3} e + a^{3} c^{7} d e^{3}\right )}}{c^{8}}\right )} x + \frac{315 \,{\left (176 \, a^{2} c^{8} d^{4} + 60 \, a^{3} c^{7} d^{2} e^{2} - 3 \, a^{4} c^{6} e^{4}\right )}}{c^{8}}\right )} x + \frac{5120 \,{\left (9 \, a^{3} c^{7} d^{3} e - 2 \, a^{4} c^{6} d e^{3}\right )}}{c^{8}}\right )} - \frac{{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{256 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(e*x + d)^4,x, algorithm="giac")

[Out]

1/80640*sqrt(c*x^2 + a)*((2*((4*((2*(7*(8*(9*c^2*x*e^4 + 40*c^2*d*e^3)*x + 27*(2
0*c^10*d^2*e^2 + 7*a*c^9*e^4)/c^8)*x + 320*(9*c^10*d^3*e + 19*a*c^9*d*e^3)/c^8)*
x + 21*(80*c^10*d^4 + 1020*a*c^9*d^2*e^2 + 93*a^2*c^8*e^4)/c^8)*x + 1920*(9*a*c^
9*d^3*e + 5*a^2*c^8*d*e^3)/c^8)*x + 105*(208*a*c^9*d^4 + 708*a^2*c^8*d^2*e^2 + 3
*a^3*c^7*e^4)/c^8)*x + 2560*(27*a^2*c^8*d^3*e + a^3*c^7*d*e^3)/c^8)*x + 315*(176
*a^2*c^8*d^4 + 60*a^3*c^7*d^2*e^2 - 3*a^4*c^6*e^4)/c^8)*x + 5120*(9*a^3*c^7*d^3*
e - 2*a^4*c^6*d*e^3)/c^8) - 1/256*(80*a^3*c^2*d^4 - 60*a^4*c*d^2*e^2 + 3*a^5*e^4
)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)