3.1451 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx\)
Optimal. Leaf size=197 \[ -\frac{2 (B d-A e)}{\sqrt{d+e x} \left (c d^2-a e^2\right )}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}} \]
[Out]
(-2*(B*d - A*e))/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((Sqrt[a]*B - A*Sqrt[c])*ArcT
anh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*(Sqrt
[c]*d - Sqrt[a]*e)^(3/2)) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTanh[(c^(1/4)*Sqrt[d + e
*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2
))
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Rubi [A] time = 0.871278, antiderivative size = 197, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16
\[ -\frac{2 (B d-A e)}{\sqrt{d+e x} \left (c d^2-a e^2\right )}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)),x]
[Out]
(-2*(B*d - A*e))/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((Sqrt[a]*B - A*Sqrt[c])*ArcT
anh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*(Sqrt
[c]*d - Sqrt[a]*e)^(3/2)) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTanh[(c^(1/4)*Sqrt[d + e
*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(1/4)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2
))
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Rubi in Sympy [A] time = 142.166, size = 223, normalized size = 1.13 \[ - \frac{2 \left (A e - B d\right )}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )} - \frac{\left (A c d - B a e - \sqrt{a} \sqrt{c} \left (A e - B d\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e + \sqrt{c} d} \left (a e^{2} - c d^{2}\right )} - \frac{\left (A c d - B a e + \sqrt{a} \sqrt{c} \left (A e - B d\right )\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e - \sqrt{c} d} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+a),x)
[Out]
-2*(A*e - B*d)/(sqrt(d + e*x)*(a*e**2 - c*d**2)) - (A*c*d - B*a*e - sqrt(a)*sqrt
(c)*(A*e - B*d))*atanh(c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e + sqrt(c)*d))/(sqrt
(a)*c**(1/4)*sqrt(sqrt(a)*e + sqrt(c)*d)*(a*e**2 - c*d**2)) - (A*c*d - B*a*e + s
qrt(a)*sqrt(c)*(A*e - B*d))*atan(c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e - sqrt(c)
*d))/(sqrt(a)*c**(1/4)*sqrt(sqrt(a)*e - sqrt(c)*d)*(a*e**2 - c*d**2))
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Mathematica [A] time = 0.646793, size = 226, normalized size = 1.15 \[ \frac{2 (A e-B d)}{\sqrt{d+e x} \left (c d^2-a e^2\right )}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right ) \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right ) \sqrt{\sqrt{a} \sqrt{c} e+c d}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^(3/2)*(a - c*x^2)),x]
[Out]
(2*(-(B*d) + A*e))/((c*d^2 - a*e^2)*Sqrt[d + e*x]) + ((Sqrt[a]*B - A*Sqrt[c])*Ar
cTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt[c]*
d - Sqrt[a]*e)*Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]) + ((Sqrt[a]*B + A*Sqrt[c])*ArcTanh
[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt[c]*d + S
qrt[a]*e)*Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])
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Maple [B] time = 0.043, size = 588, normalized size = 3. \[ -{\frac{A{c}^{2}de}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{B{e}^{2}ac}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{Ace}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{Bcd}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{A{c}^{2}de}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{B{e}^{2}ac}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{Ace}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{Bcd}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-2\,{\frac{Ae}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}}+2\,{\frac{Bd}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+a),x)
[Out]
-1/(a*e^2-c*d^2)*c^2/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(
e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d*e+1/(a*e^2-c*d^2)*c/(a*c*e^2)^
(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1
/2))*c)^(1/2))*a*B*e^2+1/(a*e^2-c*d^2)*c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh
(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*e-1/(a*e^2-c*d^2)*c/((c*d+(a
*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))
*B*d-1/(a*e^2-c*d^2)*c^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan
(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d*e+1/(a*e^2-c*d^2)*c/(a*c*
e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e
^2)^(1/2))*c)^(1/2))*a*B*e^2-1/(a*e^2-c*d^2)*c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*
arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*e+1/(a*e^2-c*d^2)*c/(
(-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c
)^(1/2))*B*d-2/(a*e^2-c*d^2)/(e*x+d)^(1/2)*A*e+2/(a*e^2-c*d^2)/(e*x+d)^(1/2)*B*d
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{B x + A}{{\left (c x^{2} - a\right )}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(B*x + A)/((c*x^2 - a)*(e*x + d)^(3/2)),x, algorithm="maxima")
[Out]
-integrate((B*x + A)/((c*x^2 - a)*(e*x + d)^(3/2)), x)
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Fricas [A] time = 2.51728, size = 8632, normalized size = 43.82 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(B*x + A)/((c*x^2 - a)*(e*x + d)^(3/2)),x, algorithm="fricas")
[Out]
1/2*((c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2
*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e
^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*
B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*
B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^
3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c
+ A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*
a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3
*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B^3*a*c^2 - A^3
*B*c^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^
2 - (B^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) + (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c
^2 + 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a
^3*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a
^4 + A^2*B*a^3*c)*e^5 + (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 +
6*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e
^7 - A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e +
3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 +
A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 1
2*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2
)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^
6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2
*e + 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (a*
c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6
- 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*
A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A
^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (
B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 +
15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^
10 + a^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)))
- (c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a
*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2
+ 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*
c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^
3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)
*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c +
A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^
4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3*d
^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B^3*a*c^2 - A^3*B
*c^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2
- (B^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) - (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c^2
+ 7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3
*c + 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4
+ A^2*B*a^3*c)*e^5 + (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6
*B*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e^7
- A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3
*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^
3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*
(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*
e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6
+ 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e
+ 2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 + (a*c^
3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 -
12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^
4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2
*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^
4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 1
5*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10
+ a^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) +
(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c
+ A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 +
3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^
4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*
a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d
^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A
^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*
c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3*d^6
- 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B^3*a*c^2 - A^3*B*c
^3)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 -
(B^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) + (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 +
7*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3*c
+ 4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 +
A^2*B*a^3*c)*e^5 - (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B
*a^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e^7 -
A*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(
3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*
B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A
*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^
6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 +
15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e +
2*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*
d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 1
2*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*
c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B
^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*
a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*
a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 +
a^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - (
c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-(6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - (B^2*a*c +
A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3
*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)
*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^
2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2
*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4
*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^
4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))/(a*c^3*d^6 -
3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((2*(A*B^3*a*c^2 - A^3*B*c^3
)*d^3 - 3*(B^4*a^2*c - A^4*c^3)*d^2*e + 6*(A*B^3*a^2*c - A^3*B*a*c^2)*d*e^2 - (B
^4*a^3 - A^4*a*c^2)*e^3)*sqrt(e*x + d) - (2*A*B^2*a*c^3*d^5 - (3*B^3*a^2*c^2 + 7
*A^2*B*a*c^3)*d^4*e + 2*(7*A*B^2*a^2*c^2 + 3*A^3*a*c^3)*d^3*e^2 - 4*(B^3*a^3*c +
4*A^2*B*a^2*c^2)*d^2*e^3 + 2*(4*A*B^2*a^3*c + A^3*a^2*c^2)*d*e^4 - (B^3*a^4 + A
^2*B*a^3*c)*e^5 - (A*a*c^5*d^8 - 2*B*a^2*c^4*d^7*e - 2*A*a^2*c^4*d^6*e^2 + 6*B*a
^3*c^3*d^5*e^3 - 6*B*a^4*c^2*d^3*e^5 + 2*A*a^4*c^2*d^2*e^6 + 2*B*a^5*c*d*e^7 - A
*a^5*c*e^8)*sqrt((4*A^2*B^2*c^4*d^6 - 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*
B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*
a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B
^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)
/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15
*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a^7*c*e^12)))*sqrt(-(6*A*B*a*c*d^2*e + 2
*A*B*a^2*e^3 - (B^2*a*c + A^2*c^2)*d^3 - 3*(B^2*a^2 + A^2*a*c)*d*e^2 - (a*c^3*d^
6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((4*A^2*B^2*c^4*d^6 - 12*
(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^
4)*d^4*e^2 - 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2
*a^2*c^2 + A^4*a*c^3)*d^2*e^4 - 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^
4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7*d^12 - 6*a^2*c^6*d^10*e^2 + 15*a^
3*c^5*d^8*e^4 - 20*a^4*c^4*d^6*e^6 + 15*a^5*c^3*d^4*e^8 - 6*a^6*c^2*d^2*e^10 + a
^7*c*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - 4*B
*d + 4*A*e)/((c*d^2 - a*e^2)*sqrt(e*x + d))
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{A}{- a d \sqrt{d + e x} - a e x \sqrt{d + e x} + c d x^{2} \sqrt{d + e x} + c e x^{3} \sqrt{d + e x}}\, dx - \int \frac{B x}{- a d \sqrt{d + e x} - a e x \sqrt{d + e x} + c d x^{2} \sqrt{d + e x} + c e x^{3} \sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+a),x)
[Out]
-Integral(A/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) +
c*e*x**3*sqrt(d + e*x)), x) - Integral(B*x/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d +
e*x) + c*d*x**2*sqrt(d + e*x) + c*e*x**3*sqrt(d + e*x)), x)
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(B*x + A)/((c*x^2 - a)*(e*x + d)^(3/2)),x, algorithm="giac")
[Out]
Timed out