3.1477 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)
Optimal. Leaf size=448 \[ -\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-28 A c d e+32 B c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 \sqrt{c} e^5 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a A e^3+29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt{d+e x}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{35 e^4} \]
[Out]
(4*Sqrt[d + e*x]*(5*a*B*e^2 + 4*c*d*(8*B*d - 7*A*e) - 3*c*e*(8*B*d - 7*A*e)*x)*S
qrt[a + c*x^2])/(35*e^4) + (2*(8*B*d - 7*A*e + B*e*x)*(a + c*x^2)^(3/2))/(7*e^2*
Sqrt[d + e*x]) + (8*Sqrt[-a]*Sqrt[c]*(32*B*c*d^3 - 28*A*c*d^2*e + 29*a*B*d*e^2 -
21*a*A*e^3)*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)])/(35*e^5*Sqrt[(Sqr
t[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2
+ a*e^2)*(32*B*c*d^2 - 28*A*c*d*e + 5*a*B*e^2)*Sqrt[(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)])/(35*Sqrt[c]*e^5*Sqrt[d + e
*x]*Sqrt[a + c*x^2])
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Rubi [A] time = 1.13328, antiderivative size = 448, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231
\[ -\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-28 A c d e+32 B c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 \sqrt{c} e^5 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a A e^3+29 a B d e^2-28 A c d^2 e+32 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \left (a+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt{d+e x}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (5 a B e^2-3 c e x (8 B d-7 A e)+4 c d (8 B d-7 A e)\right )}{35 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^(3/2))/(d + e*x)^(3/2),x]
[Out]
(4*Sqrt[d + e*x]*(5*a*B*e^2 + 4*c*d*(8*B*d - 7*A*e) - 3*c*e*(8*B*d - 7*A*e)*x)*S
qrt[a + c*x^2])/(35*e^4) + (2*(8*B*d - 7*A*e + B*e*x)*(a + c*x^2)^(3/2))/(7*e^2*
Sqrt[d + e*x]) + (8*Sqrt[-a]*Sqrt[c]*(32*B*c*d^3 - 28*A*c*d^2*e + 29*a*B*d*e^2 -
21*a*A*e^3)*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)])/(35*e^5*Sqrt[(Sqr
t[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (8*Sqrt[-a]*(c*d^2
+ a*e^2)*(32*B*c*d^2 - 28*A*c*d*e + 5*a*B*e^2)*Sqrt[(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)])/(35*Sqrt[c]*e^5*Sqrt[d + e
*x]*Sqrt[a + c*x^2])
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Rubi in Sympy [A] time = 172.89, size = 454, normalized size = 1.01 \[ - \frac{8 \sqrt{c} \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (21 A a e^{3} + 28 A c d^{2} e - 29 B a d e^{2} - 32 B c d^{3}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{35 e^{5} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} - \frac{4 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (\frac{7 A e}{2} - 4 B d - \frac{B e x}{2}\right )}{7 e^{2} \sqrt{d + e x}} + \frac{8 \sqrt{a + c x^{2}} \sqrt{d + e x} \left (\frac{5 B a e^{2}}{2} - 2 c d \left (7 A e - 8 B d\right ) + \frac{3 c e x \left (7 A e - 8 B d\right )}{2}\right )}{35 e^{4}} - \frac{8 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + c d^{2}\right ) \left (- 28 A c d e + 5 B a e^{2} + 32 B c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{35 \sqrt{c} e^{5} \sqrt{a + c x^{2}} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/(e*x+d)**(3/2),x)
[Out]
-8*sqrt(c)*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(21*A*a*e**3 + 28*A*c*d**2*
e - 29*B*a*d*e**2 - 32*B*c*d**3)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) +
1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(35*e**5*sqrt(sqrt(c)*sqrt(-a)*(-d - e*
x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)) - 4*(a + c*x**2)**(3/2)*(7*A*e/
2 - 4*B*d - B*e*x/2)/(7*e**2*sqrt(d + e*x)) + 8*sqrt(a + c*x**2)*sqrt(d + e*x)*(
5*B*a*e**2/2 - 2*c*d*(7*A*e - 8*B*d) + 3*c*e*x*(7*A*e - 8*B*d)/2)/(35*e**4) - 8*
sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c
*x**2/a)*(a*e**2 + c*d**2)*(-28*A*c*d*e + 5*B*a*e**2 + 32*B*c*d**2)*elliptic_f(a
sin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(35*
sqrt(c)*e**5*sqrt(a + c*x**2)*sqrt(d + e*x))
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Mathematica [C] time = 8.40324, size = 661, normalized size = 1.48 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (B \left (5 a e^2 (10 d+3 e x)+c \left (64 d^3+16 d^2 e x-8 d e^2 x^2+5 e^3 x^3\right )\right )-7 A e \left (5 a e^2+c \left (8 d^2+2 d e x-e^2 x^2\right )\right )\right )}{e^4 (d+e x)}+\frac{8 \left (\sqrt{a} e (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (21 i \sqrt{a} A \sqrt{c} e^2-24 i \sqrt{a} B \sqrt{c} d e+5 a B e^2-28 A c d e+32 B c d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (21 a A e^3-29 a B d e^2+28 A c d^2 e-32 B c d^3\right )+\sqrt{c} (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (21 a A e^3-29 a B d e^2+28 A c d^2 e-32 B c d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{e^6 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{35 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^(3/2))/(d + e*x)^(3/2),x]
[Out]
(Sqrt[d + e*x]*((2*(a + c*x^2)*(-7*A*e*(5*a*e^2 + c*(8*d^2 + 2*d*e*x - e^2*x^2))
+ B*(5*a*e^2*(10*d + 3*e*x) + c*(64*d^3 + 16*d^2*e*x - 8*d*e^2*x^2 + 5*e^3*x^3)
)))/(e^4*(d + e*x)) + (8*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-32*B*c*d^3 + 28
*A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*(a + c*x^2) + Sqrt[c]*((-I)*Sqrt[c]*d +
Sqrt[a]*e)*(-32*B*c*d^3 + 28*A*c*d^2*e - 29*a*B*d*e^2 + 21*a*A*e^3)*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]]/Sqrt[d +
e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(Sqrt[c
]*d + I*Sqrt[a]*e)*(32*B*c*d^2 - (24*I)*Sqrt[a]*B*Sqrt[c]*d*e - 28*A*c*d*e + 5*a
*B*e^2 + (21*I)*Sqrt[a]*A*Sqrt[c]*e^2)*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)*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)]))/(e^6*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x
))))/(35*Sqrt[a + c*x^2])
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Maple [B] time = 0.047, size = 2561, normalized size = 5.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(3/2)/(e*x+d)^(3/2),x)
[Out]
-2/35*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(-5*B*x^5*c^3*e^5-16*B*x*a*c^2*d^2*e^3-42*B*
x^2*a*c^2*d*e^4+14*A*x*a*c^2*d*e^4-116*B*(-(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^2*c*d*e^4-244*B*(-(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*c^2*d^3*e^2-1
5*B*x*a^2*c*e^5+8*B*x^4*c^3*d*e^4+14*A*x^3*c^3*d*e^4-20*B*x^3*a*c^2*e^5+56*A*a*c
^2*d^2*e^3-50*B*a^2*c*d*e^4-64*B*a*c^2*d^3*e^2+128*B*(-(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*c)^(1/2)*c^2*d^4*e-8
4*A*(-(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*c^2*d^2*e^3-112*A*(-(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*c)^(1/2)*c^2*d^3*e^2+196*A*(-(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*c^2*d^2*e^3+96
*B*(-(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^2*c*d*e^4+96*B*(-(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*c^2*d^3*e^2-128*B*(-(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))*c^3*d^5+112*A*(-(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))*c^3*d^4*e+20*B
*(-(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*c)^(1/2)*a^2*e^5-84*A*(-(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^2*c*e^5-64*B*x^2*c^3*d^3*e^2+148*B*(-(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*c)^
(1/2)*a*c*d^2*e^3-112*A*(-(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*c)^(1/2)*a*c*d*e^4+35*A*a^2*c*e^5-7*A*x^4*c^3*e^5
-16*B*x^3*c^3*d^2*e^3+28*A*x^2*a*c^2*e^5+56*A*x^2*c^3*d^2*e^3+84*A*(-(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^2*c*e^5)
/c/e^6/(c*e*x^3+c*d*x^2+a*e*x+a*d)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
integrate((c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(3/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt{c x^{2} + a}}{{\left (e x + d\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
integral((B*c*x^3 + A*c*x^2 + B*a*x + A*a)*sqrt(c*x^2 + a)/(e*x + d)^(3/2), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(3/2)/(e*x+d)**(3/2),x)
[Out]
Integral((A + B*x)*(a + c*x**2)**(3/2)/(d + e*x)**(3/2), x)
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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)^(3/2)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError