3.601 \(\int \frac{(d+e x)^{5/2}}{a-c x^2} \, dx\)

Optimal. Leaf size=167 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{7/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{7/4}}-\frac{2 e (d+e x)^{3/2}}{3 c}-\frac{4 d e \sqrt{d+e x}}{c} \]

[Out]

(-4*d*e*Sqrt[d + e*x])/c - (2*e*(d + e*x)^(3/2))/(3*c) - ((Sqrt[c]*d - Sqrt[a]*e
)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c
^(7/4)) + ((Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sq
rt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(7/4))

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Rubi [A]  time = 0.951847, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{7/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{7/4}}-\frac{2 e (d+e x)^{3/2}}{3 c}-\frac{4 d e \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

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

[Out]

(-4*d*e*Sqrt[d + e*x])/c - (2*e*(d + e*x)^(3/2))/(3*c) - ((Sqrt[c]*d - Sqrt[a]*e
)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c
^(7/4)) + ((Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sq
rt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(7/4))

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Rubi in Sympy [A]  time = 171.987, size = 236, normalized size = 1.41 \[ - \frac{4 d e \sqrt{d + e x}}{c} - \frac{2 e \left (d + e x\right )^{\frac{3}{2}}}{3 c} + \frac{\left (a^{\frac{3}{2}} e^{3} + 3 \sqrt{a} c d^{2} e + 3 a \sqrt{c} d e^{2} + c^{\frac{3}{2}} d^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{7}{4}} \sqrt{\sqrt{a} e + \sqrt{c} d}} - \frac{\left (a^{\frac{3}{2}} e^{3} + 3 \sqrt{a} c d^{2} e - 3 a \sqrt{c} d e^{2} - c^{\frac{3}{2}} d^{3}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{7}{4}} \sqrt{\sqrt{a} e - \sqrt{c} d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*d*e*sqrt(d + e*x)/c - 2*e*(d + e*x)**(3/2)/(3*c) + (a**(3/2)*e**3 + 3*sqrt(a)
*c*d**2*e + 3*a*sqrt(c)*d*e**2 + c**(3/2)*d**3)*atanh(c**(1/4)*sqrt(d + e*x)/sqr
t(sqrt(a)*e + sqrt(c)*d))/(sqrt(a)*c**(7/4)*sqrt(sqrt(a)*e + sqrt(c)*d)) - (a**(
3/2)*e**3 + 3*sqrt(a)*c*d**2*e - 3*a*sqrt(c)*d*e**2 - c**(3/2)*d**3)*atan(c**(1/
4)*sqrt(d + e*x)/sqrt(sqrt(a)*e - sqrt(c)*d))/(sqrt(a)*c**(7/4)*sqrt(sqrt(a)*e -
sqrt(c)*d))

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Mathematica [A]  time = 0.265442, size = 197, normalized size = 1.18 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c^{3/2} \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} c^{3/2} \sqrt{\sqrt{a} \sqrt{c} e+c d}}-\frac{2 e \sqrt{d+e x} (7 d+e x)}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*Sqrt[d + e*x]*(7*d + e*x))/(3*c) - ((Sqrt[c]*d - Sqrt[a]*e)^3*ArcTanh[(Sqr
t[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c^(3/2)*Sqrt[c*d -
Sqrt[a]*Sqrt[c]*e]) + ((Sqrt[c]*d + Sqrt[a]*e)^3*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c^(3/2)*Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])

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Maple [B]  time = 0.072, size = 460, normalized size = 2.8 \[ -{\frac{2\,e}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{de\sqrt{ex+d}}{c}}+3\,{\frac{ad{e}^{3}}{\sqrt{ac{e}^{2}}\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) }+{ce{d}^{3}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{a{e}^{3}}{c}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+3\,{\frac{e{d}^{2}}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) }+3\,{\frac{ad{e}^{3}}{\sqrt{ac{e}^{2}}\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) }+{ce{d}^{3}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{a{e}^{3}}{c}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-3\,{\frac{e{d}^{2}}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(-c*x^2+a),x)

[Out]

-2/3*e*(e*x+d)^(3/2)/c-4*d*e*(e*x+d)^(1/2)/c+3/(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^3+
e*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))*d^3+1/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^3+3*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^2+3/(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^3+e*c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*ar
ctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^3-1/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^3-3
*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^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.298415, size = 2183, normalized size = 13.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25*c^4*d^8*
e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*
c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^
9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a
*c^6*d^2 + a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d
^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 +
5*a^2*d*e^4 + a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*
e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt((c^2*d^5 + 10*
a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110
*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^
8*e - 14*a^2*c^2*d^4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) - (10*a*c^4*
d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 - (a*c^6*d^2 + a^2*c^5*e^2)*sqrt(
(25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a
^4*e^10)/(a*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt((25
*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*
e^10)/(a*c^7)))/(a*c^3))) + 3*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a
*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d
^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*
a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^4
+ 2*a^3*c^2*d*e^6 + (a*c^6*d^2 + a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d
^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt((c^2*
d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*
e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) - 3
*c*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 1
00*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/
(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 + 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt
(e*x + d) - (10*a*c^4*d^5*e^2 + 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^
2 + a^2*c^5*e^2)*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6
+ 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt((c^2*d^5 + 10*a*c*d^3*e^2 + 5*a^2*
d*e^4 - a*c^3*sqrt((25*c^4*d^8*e^2 + 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 + 2
0*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) + 4*(e^2*x + 7*d*e)*sqrt(e*x + d
))/c

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Sympy [A]  time = 121.382, size = 418, normalized size = 2.5 \[ - \frac{4 a d e^{3} \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )}}{c} - \frac{2 a e^{3} \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )}}{c} + 4 d^{3} e \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log{\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt{d + e x} \right )} \right )\right )} - 6 d^{2} e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} - \frac{4 d e \sqrt{d + e x}}{c} - \frac{2 e \left (d + e x\right )^{\frac{3}{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a*d*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*a*e**3*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*lo
g(-64*_t**3*a*c**2*e**2 + 4*_t*c*d + sqrt(d + e*x))))/c + 4*d**3*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)))) - 6*d**2*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)))) - 4*d*e*sqrt(d + e*x)/c - 2*e*(d + e*x)**(3/2)/(3*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)^(5/2)/(c*x^2 - a),x, algorithm="giac")

[Out]

Exception raised: TypeError