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

Optimal. Leaf size=566 \[ -\frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (d \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right )-3 e x \left (-77 a^2 e^4+27 a c d^2 e^2+8 c^2 d^4\right )\right )}{9009 e^5}+\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-231 a^3 e^6+258 a^2 c d^2 e^4+137 a c^2 d^4 e^2+32 c^3 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (4 d \left (5 a e^2+2 c d^2\right )-7 e x \left (c d^2-11 a e^2\right )\right )}{9009 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{20 d \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{143 e} \]

[Out]

(8*Sqrt[d + e*x]*(d*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4) - 3*e*(8*c^2*d^
4 + 27*a*c*d^2*e^2 - 77*a^2*e^4)*x)*Sqrt[a + c*x^2])/(9009*e^5) + (20*Sqrt[d + e
*x]*(4*d*(2*c*d^2 + 5*a*e^2) - 7*e*(c*d^2 - 11*a*e^2)*x)*(a + c*x^2)^(3/2))/(900
9*e^3) - (20*d*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(143*e) + (2*(d + e*x)^(3/2)*(a
+ c*x^2)^(5/2))/(13*e) + (16*Sqrt[-a]*(32*c^3*d^6 + 137*a*c^2*d^4*e^2 + 258*a^2*
c*d^2*e^4 - 231*a^3*e^6)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt
[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(9009
*Sqrt[c]*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2])
- (16*Sqrt[-a]*d*(c*d^2 + a*e^2)*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4)*S
qrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[
ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a
*e)])/(9009*Sqrt[c]*e^6*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.72553, antiderivative size = 566, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (d \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right )-3 e x \left (-77 a^2 e^4+27 a c d^2 e^2+8 c^2 d^4\right )\right )}{9009 e^5}+\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-231 a^3 e^6+258 a^2 c d^2 e^4+137 a c^2 d^4 e^2+32 c^3 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{9009 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (4 d \left (5 a e^2+2 c d^2\right )-7 e x \left (c d^2-11 a e^2\right )\right )}{9009 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{20 d \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{143 e} \]

Antiderivative was successfully verified.

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

[Out]

(8*Sqrt[d + e*x]*(d*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4) - 3*e*(8*c^2*d^
4 + 27*a*c*d^2*e^2 - 77*a^2*e^4)*x)*Sqrt[a + c*x^2])/(9009*e^5) + (20*Sqrt[d + e
*x]*(4*d*(2*c*d^2 + 5*a*e^2) - 7*e*(c*d^2 - 11*a*e^2)*x)*(a + c*x^2)^(3/2))/(900
9*e^3) - (20*d*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(143*e) + (2*(d + e*x)^(3/2)*(a
+ c*x^2)^(5/2))/(13*e) + (16*Sqrt[-a]*(32*c^3*d^6 + 137*a*c^2*d^4*e^2 + 258*a^2*
c*d^2*e^4 - 231*a^3*e^6)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt
[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(9009
*Sqrt[c]*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2])
- (16*Sqrt[-a]*d*(c*d^2 + a*e^2)*(32*c^2*d^4 + 113*a*c*d^2*e^2 + 177*a^2*e^4)*S
qrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[
ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a
*e)])/(9009*Sqrt[c]*e^6*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 8.9351, size = 790, normalized size = 1.4 \[ \frac{2 \sqrt{d+e x} \left (e^2 \left (a+c x^2\right ) \left (a^2 e^4 (971 d+2387 e x)+2 a c e^2 \left (266 d^3-197 d^2 e x+163 d e^2 x^2+1078 e^3 x^3\right )+c^2 \left (128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right )\right )+\frac{8 \left (e^2 \left (a+c x^2\right ) \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (-231 a^3 e^6+258 a^2 c d^2 e^4+137 a c^2 d^4 e^2+32 c^3 d^6\right )+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (32 i a^{3/2} c^{3/2} d^3 e^3+408 i a^{5/2} \sqrt{c} d e^5-231 a^3 e^6+258 a^2 c d^2 e^4+8 i \sqrt{a} c^{5/2} d^5 e+137 a c^2 d^4 e^2+32 c^3 d^6\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+\sqrt{c} (d+e x)^{3/2} \left (-137 a^{3/2} c^2 d^4 e^3-258 a^{5/2} c d^2 e^5+231 a^{7/2} e^7-231 i a^3 \sqrt{c} d e^6+258 i a^2 c^{3/2} d^3 e^4+137 i a c^{5/2} d^5 e^2-32 \sqrt{a} c^3 d^6 e+32 i c^{7/2} d^7\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{9009 e^7 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(e^2*(a + c*x^2)*(a^2*e^4*(971*d + 2387*e*x) + 2*a*c*e^2*(266*d
^3 - 197*d^2*e*x + 163*d*e^2*x^2 + 1078*e^3*x^3) + c^2*(128*d^5 - 96*d^4*e*x + 8
0*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)) + (8*(-(e^2*Sqrt[-
d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^3*d^6 + 137*a*c^2*d^4*e^2 + 258*a^2*c*d^2*e^4 -
231*a^3*e^6)*(a + c*x^2)) + Sqrt[c]*((32*I)*c^(7/2)*d^7 - 32*Sqrt[a]*c^3*d^6*e
+ (137*I)*a*c^(5/2)*d^5*e^2 - 137*a^(3/2)*c^2*d^4*e^3 + (258*I)*a^2*c^(3/2)*d^3*
e^4 - 258*a^(5/2)*c*d^2*e^5 - (231*I)*a^3*Sqrt[c]*d*e^6 + 231*a^(7/2)*e^7)*Sqrt[
(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d
+ e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/S
qrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sq
rt[c]*e*(32*c^3*d^6 + (8*I)*Sqrt[a]*c^(5/2)*d^5*e + 137*a*c^2*d^4*e^2 + (32*I)*a
^(3/2)*c^(3/2)*d^3*e^3 + 258*a^2*c*d^2*e^4 + (408*I)*a^(5/2)*Sqrt[c]*d*e^5 - 231
*a^3*e^6)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sq
rt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a
]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e
)]))/(c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(9009*e^7*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.054, size = 2332, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)*(e*x+d)^(1/2),x)

[Out]

2/9009*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(971*a^3*c*d^2*e^6+532*a^2*c^2*d^4*e^4+128*
a*c^3*d^6*e^2-1848*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*
e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*
EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)
^(1/2)*e+c*d))^(1/2))*a^4*e^8-2320*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(
1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d^3*e^5*(-a*c)^(1
/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)
*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-1160*EllipticF(
(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c
*d))^(1/2))*a*c^2*d^5*e^3*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*(
(-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^
(1/2)*e-c*d))^(1/2)+3160*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-
a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^4*e^4*(-(e*x+d)*c/((-a*
c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+
(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+1352*EllipticE((-(e*x+d)*c/((-a*c)^(
1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^3*d^6
*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2
)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-840*EllipticF(
(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c
*d))^(1/2))*a^2*c^2*d^4*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d
))^(1/2)+256*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e
-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^4*d^8*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/
2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e-c*d))^(1/2)+693*x^8*c^4*e^8+756*x^7*c^4*d*e^7+2849*x^6*a*c^3*e^8-7*x
^6*c^4*d^2*e^6-75*x^4*a*c^3*d^2*e^6+5840*x^3*a^2*c^2*d*e^7+148*x^3*a*c^3*d^3*e^5
+903*x^2*a^2*c^2*d^2*e^6+516*x^2*a*c^3*d^4*e^4+3358*x*a^3*c*d*e^7+138*x*a^2*c^2*
d^3*e^5+32*x*a*c^3*d^5*e^3+3238*x^5*a*c^3*d*e^7+1848*(-(e*x+d)*c/((-a*c)^(1/2)*e
-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/
2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^4*e^8-192*EllipticF((-
(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d
))^(1/2))*a*c^3*d^6*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1
/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(
1/2)-256*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d
)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^3*d^7*e*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*
e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1
/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+10*x^5*c^4*d^3*e^5+4543*x^4*a^2*c^2*e^8-16*x^
4*c^4*d^4*e^4+32*x^3*c^4*d^5*e^3+2387*x^2*a^3*c*e^8+128*x^2*c^4*d^6*e^2+216*(-(e
*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))
^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a
^3*c*d^2*e^6-1416*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e
/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*E
llipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^
(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*a^3*d*e^7+1200*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d
))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*
e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),
(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d^2*e^6)/c/(c*e*x^3+c*
d*x^2+a*e*x+a*d)/e^7

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{\frac{5}{2}} \sqrt{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + a)^(5/2)*sqrt(e*x + d), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)*sqrt(e*x + d), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + c x^{2}\right )^{\frac{5}{2}} \sqrt{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Integral((a + c*x**2)**(5/2)*sqrt(d + e*x), x)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError