∖int A+Bx___________√ d+ex ∖left(a−cx2∖right)∖,dx

3.1450 \(\int \frac{A+B x}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx\)

Optimal. Leaf size=152 \[ \frac{\left (B-\frac{A \sqrt{c}}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{\left (\frac{A \sqrt{c}}{\sqrt{a}}+B\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{c^{3/4} \sqrt{\sqrt{a} e+\sqrt{c} d}} \]

[Out]

((B - (A*Sqrt[c])/Sqrt[a])*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt

[a]*e]])/(c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((B + (A*Sqrt[c])/Sqrt[a])*ArcT

anh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(c^(3/4)*Sqrt[Sqrt[c]*

d + Sqrt[a]*e])

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Rubi [A] time = 0.334659, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{\left (B-\frac{A \sqrt{c}}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{\left (\frac{A \sqrt{c}}{\sqrt{a}}+B\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{c^{3/4} \sqrt{\sqrt{a} e+\sqrt{c} d}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

((B - (A*Sqrt[c])/Sqrt[a])*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt

[a]*e]])/(c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((B + (A*Sqrt[c])/Sqrt[a])*ArcT

anh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(c^(3/4)*Sqrt[Sqrt[c]*

d + Sqrt[a]*e])

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Rubi in Sympy [A] time = 63.3268, size = 144, normalized size = 0.95 \[ \frac{\left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{3}{4}} \sqrt{\sqrt{a} e - \sqrt{c} d}} + \frac{\left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{3}{4}} \sqrt{\sqrt{a} e + \sqrt{c} d}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*sqrt(c) - B*sqrt(a))*atan(c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e - sqrt(c)*d))

/(sqrt(a)*c**(3/4)*sqrt(sqrt(a)*e - sqrt(c)*d)) + (A*sqrt(c) + B*sqrt(a))*atanh(

c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e + sqrt(c)*d))/(sqrt(a)*c**(3/4)*sqrt(sqrt(

a)*e + sqrt(c)*d))

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Mathematica [A] time = 0.354509, size = 159, normalized size = 1.05 \[ \frac{\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}}{\sqrt{a} \sqrt{c}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(((Sqrt[a]*B - A*Sqrt[c])*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqr

t[c]*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[c*d + Sqrt[a]*Sqrt[c]*e]

)/(Sqrt[a]*Sqrt[c])

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Maple [A] time = 0.036, size = 203, normalized size = 1.3 \[{Ace{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{B{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{Ace\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{B\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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*e+1/((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+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*e-1/((

-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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((B*x + A)/((c*x^2 - a)*sqrt(e*x + d)), x)

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Fricas [A] time = 0.326475, size = 3220, normalized size = 21.18 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)*d + (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B

^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)

*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))*l

og((2*(A*B^3*a*c - A^3*B*c^2)*d - (B^4*a^2 - A^4*c^2)*e)*sqrt(e*x + d) + (2*A*B^

2*a*c^2*d^2 - (B^3*a^2*c + 3*A^2*B*a*c^2)*d*e + (A*B^2*a^2*c + A^3*a*c^2)*e^2 +

(A*a*c^4*d^3 - B*a^2*c^3*d^2*e - A*a^2*c^3*d*e^2 + B*a^3*c^2*e^3)*sqrt((4*A^2*B^

2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*

e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))*sqrt(-(2*A*B*a*e - (B^2*a +

A^2*c)*d + (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3

*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^

2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))) - 1/2*sqrt(-(2*A*B*a*e - (B^2*a

+ A^2*c)*d + (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A

^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*

d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))*log((2*(A*B^3*a*c - A^3*B*c^2)

*d - (B^4*a^2 - A^4*c^2)*e)*sqrt(e*x + d) - (2*A*B^2*a*c^2*d^2 - (B^3*a^2*c + 3*

A^2*B*a*c^2)*d*e + (A*B^2*a^2*c + A^3*a*c^2)*e^2 + (A*a*c^4*d^3 - B*a^2*c^3*d^2*

e - A*a^2*c^3*d*e^2 + B*a^3*c^2*e^3)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^

3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d

^2*e^2 + a^3*c^3*e^4)))*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)*d + (a*c^2*d^2 - a^2*

c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^

2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2

*d^2 - a^2*c*e^2))) + 1/2*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)*d - (a*c^2*d^2 - a^

2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*

A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c

^2*d^2 - a^2*c*e^2))*log((2*(A*B^3*a*c - A^3*B*c^2)*d - (B^4*a^2 - A^4*c^2)*e)*s

qrt(e*x + d) + (2*A*B^2*a*c^2*d^2 - (B^3*a^2*c + 3*A^2*B*a*c^2)*d*e + (A*B^2*a^2

*c + A^3*a*c^2)*e^2 - (A*a*c^4*d^3 - B*a^2*c^3*d^2*e - A*a^2*c^3*d*e^2 + B*a^3*c

^2*e^3)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A

^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))*sqrt(

-(2*A*B*a*e - (B^2*a + A^2*c)*d - (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^

2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*

c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))) - 1/2*sqr

t(-(2*A*B*a*e - (B^2*a + A^2*c)*d - (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B^2*c^2*

d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(

a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))*log((2*(

A*B^3*a*c - A^3*B*c^2)*d - (B^4*a^2 - A^4*c^2)*e)*sqrt(e*x + d) - (2*A*B^2*a*c^2

*d^2 - (B^3*a^2*c + 3*A^2*B*a*c^2)*d*e + (A*B^2*a^2*c + A^3*a*c^2)*e^2 - (A*a*c^

4*d^3 - B*a^2*c^3*d^2*e - A*a^2*c^3*d*e^2 + B*a^3*c^2*e^3)*sqrt((4*A^2*B^2*c^2*d

^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a

*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)

*d - (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)

*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 +

a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2)))

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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{A}{- a \sqrt{d + e x} + c x^{2} \sqrt{d + e x}}\, dx - \int \frac{B x}{- a \sqrt{d + e x} + c x^{2} \sqrt{d + e x}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(A/(-a*sqrt(d + e*x) + c*x**2*sqrt(d + e*x)), x) - Integral(B*x/(-a*sqr

t(d + e*x) + c*x**2*sqrt(d + e*x)), x)

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GIAC/XCAS [A] time = 100.427, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Done

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