3.602 \(\int \frac{(d+e x)^{3/2}}{a-c x^2} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{5/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{5/4}}-\frac{2 e \sqrt{d+e x}}{c} \]

[Out]

(-2*e*Sqrt[d + e*x])/c - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(5/4)) + ((Sqrt[c]*d + Sqrt[a]*
e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*
c^(5/4))

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Rubi [A]  time = 0.737005, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{5/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{5/4}}-\frac{2 e \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)/(a - c*x^2),x]

[Out]

(-2*e*Sqrt[d + e*x])/c - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(5/4)) + ((Sqrt[c]*d + Sqrt[a]*
e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*
c^(5/4))

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Rubi in Sympy [A]  time = 130.611, size = 182, normalized size = 1.22 \[ - \frac{2 e \sqrt{d + e x}}{c} + \frac{\left (2 \sqrt{a} \sqrt{c} d e + a e^{2} + c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{5}{4}} \sqrt{\sqrt{a} e + \sqrt{c} d}} + \frac{\left (- 2 \sqrt{a} \sqrt{c} d e + a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{5}{4}} \sqrt{\sqrt{a} e - \sqrt{c} d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)/(-c*x**2+a),x)

[Out]

-2*e*sqrt(d + e*x)/c + (2*sqrt(a)*sqrt(c)*d*e + a*e**2 + c*d**2)*atanh(c**(1/4)*
sqrt(d + e*x)/sqrt(sqrt(a)*e + sqrt(c)*d))/(sqrt(a)*c**(5/4)*sqrt(sqrt(a)*e + sq
rt(c)*d)) + (-2*sqrt(a)*sqrt(c)*d*e + a*e**2 + c*d**2)*atan(c**(1/4)*sqrt(d + e*
x)/sqrt(sqrt(a)*e - sqrt(c)*d))/(sqrt(a)*c**(5/4)*sqrt(sqrt(a)*e - sqrt(c)*d))

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Mathematica [A]  time = 0.169764, size = 184, normalized size = 1.23 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} c \sqrt{\sqrt{a} \sqrt{c} e+c d}}-\frac{2 e \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)/(a - c*x^2),x]

[Out]

(-2*e*Sqrt[d + e*x])/c - ((Sqrt[c]*d - Sqrt[a]*e)^2*ArcTanh[(Sqrt[c]*Sqrt[d + e*
x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c*Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]) +
((Sqrt[c]*d + Sqrt[a]*e)^2*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sq
rt[c]*e]])/(Sqrt[a]*c*Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])

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Maple [B]  time = 0.034, size = 335, normalized size = 2.3 \[ -2\,{\frac{e\sqrt{ex+d}}{c}}+{a{e}^{3}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ce{d}^{2}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+2\,{\frac{de}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) }+{a{e}^{3}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ce{d}^{2}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-2\,{\frac{de}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/(-c*x^2+a),x)

[Out]

-2*e*(e*x+d)^(1/2)/c+1/(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^3+e/(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))
*c*d^2+2*e/((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))*d+1/(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^3+e/(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))*c*d^2-2*e/((-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))*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{c x^{2} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^(3/2)/(c*x^2 - a),x, algorithm="maxima")

[Out]

-integrate((e*x + d)^(3/2)/(c*x^2 - a), x)

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Fricas [A]  time = 0.252992, size = 1315, normalized size = 8.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^(3/2)/(c*x^2 - a),x, algorithm="fricas")

[Out]

1/2*(c*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2
*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x +
d) + (3*a*c^2*d^2*e^2 + a^2*c*e^4 - a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4
+ a^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*
c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt((c*d^3 + 3*a*d*e^2 + a*c^2*sqr
t((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e
- 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 + a^2*c*e^4 - a*c^4
*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*
e^2 + a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) +
c*sqrt((c*d^3 + 3*a*d*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6
)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d)
+ (3*a*c^2*d^2*e^2 + a^2*c*e^4 + a*c^4*d*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a
^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2 - a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^
2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - c*sqrt((c*d^3 + 3*a*d*e^2 - a*c^2*sqrt((9
*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))*log(-(3*c^2*d^4*e - 2
*a*c*d^2*e^3 - a^2*e^5)*sqrt(e*x + d) - (3*a*c^2*d^2*e^2 + a^2*c*e^4 + a*c^4*d*s
qrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))*sqrt((c*d^3 + 3*a*d*e^2
- a*c^2*sqrt((9*c^2*d^4*e^2 + 6*a*c*d^2*e^4 + a^2*e^6)/(a*c^5)))/(a*c^2))) - 4*s
qrt(e*x + d)*e)/c

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Sympy [A]  time = 67.4941, size = 316, normalized size = 2.12 \[ - \frac{2 a e^{3} \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )}}{c} + 2 d^{2} e \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )} - 4 d e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} - \frac{2 e \sqrt{d + e x}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)/(-c*x**2+a),x)

[Out]

-2*a*e**3*RootSum(_t**4*(256*a**3*c*e**6 - 256*a**2*c**2*d**2*e**4) + 32*_t**2*a
*c*d*e**2 - 1, Lambda(_t, _t*log(-64*_t**3*a**2*c*d*e**4 + 64*_t**3*a*c**2*d**3*
e**2 - 4*_t*a*e**2 - 4*_t*c*d**2 + sqrt(d + e*x))))/c + 2*d**2*e*RootSum(_t**4*(
256*a**3*c*e**6 - 256*a**2*c**2*d**2*e**4) + 32*_t**2*a*c*d*e**2 - 1, Lambda(_t,
_t*log(-64*_t**3*a**2*c*d*e**4 + 64*_t**3*a*c**2*d**3*e**2 - 4*_t*a*e**2 - 4*_t
*c*d**2 + sqrt(d + e*x)))) - 4*d*e*RootSum(256*_t**4*a**2*c**3*e**4 - 32*_t**2*a
*c**2*d*e**2 - a*e**2 + c*d**2, Lambda(_t, _t*log(-64*_t**3*a*c**2*e**2 + 4*_t*c
*d + sqrt(d + e*x)))) - 2*e*sqrt(d + e*x)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^(3/2)/(c*x^2 - a),x, algorithm="giac")

[Out]

Exception raised: TypeError