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

Optimal. Leaf size=209 \[ -\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \left (\sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\left (2 \sqrt{c} d-\sqrt{a} e\right ) \sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\sqrt{d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.67898, antiderivative size = 209, 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 \[ -\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \left (\sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\left (2 \sqrt{c} d-\sqrt{a} e\right ) \sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\sqrt{d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 148.502, size = 201, normalized size = 0.96 \[ \frac{\sqrt{d + e x} \left (a e + c d x\right )}{2 a c \left (a - c x^{2}\right )} - \frac{\left (- \sqrt{a} \sqrt{c} d e + a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{5}{4}} \sqrt{\sqrt{a} e + \sqrt{c} d}} - \frac{\left (\sqrt{a} \sqrt{c} d e + a e^{2} - 2 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{5}{4}} \sqrt{\sqrt{a} e - \sqrt{c} d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(d + e*x)*(a*e + c*d*x)/(2*a*c*(a - c*x**2)) - (-sqrt(a)*sqrt(c)*d*e + a*e**
2 - 2*c*d**2)*atanh(c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e + sqrt(c)*d))/(4*a**(3
/2)*c**(5/4)*sqrt(sqrt(a)*e + sqrt(c)*d)) - (sqrt(a)*sqrt(c)*d*e + a*e**2 - 2*c*
d**2)*atan(c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e - sqrt(c)*d))/(4*a**(3/2)*c**(5
/4)*sqrt(sqrt(a)*e - sqrt(c)*d))

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Mathematica [A]  time = 0.335529, size = 218, normalized size = 1.04 \[ \frac{-\frac{\left (-\sqrt{a} \sqrt{c} d e-a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} \sqrt{c} d e-a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}-\frac{2 \sqrt{a} \sqrt{d+e x} (a e+c d x)}{c x^2-a}}{4 a^{3/2} c} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[a]*(a*e + c*d*x)*Sqrt[d + e*x])/(-a + c*x^2) - ((2*c*d^2 - Sqrt[a]*Sqr
t[c]*d*e - a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]]
)/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e] + ((2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTa
nh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt[a]*Sq
rt[c]*e])/(4*a^(3/2)*c)

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Maple [B]  time = 0.049, size = 590, normalized size = 2.8 \[ -{\frac{de}{ \left ( 2\,c{e}^{2}{x}^{2}-2\,a{e}^{2} \right ) a} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{3}}{ \left ( 2\,c{e}^{2}{x}^{2}-2\,a{e}^{2} \right ) c}\sqrt{ex+d}}+{\frac{e{d}^{2}}{ \left ( 2\,c{e}^{2}{x}^{2}-2\,a{e}^{2} \right ) a}\sqrt{ex+d}}-{\frac{{e}^{6}{a}^{2}c}{4}{\it Artanh} \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}+{\frac{{e}^{4}a{c}^{2}{d}^{2}}{2}{\it Artanh} \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}+{\frac{d{e}^{2}}{4}{\it Artanh} \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}-{\frac{{e}^{6}{a}^{2}c}{4}\arctan \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}+{\frac{{e}^{4}a{c}^{2}{d}^{2}}{2}\arctan \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}-{\frac{d{e}^{2}}{4}\arctan \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e/(c*e^2*x^2-a*e^2)*d/a*(e*x+d)^(3/2)-1/2*e^3/(c*e^2*x^2-a*e^2)/c*(e*x+d)^(
1/2)+1/2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*d^2-1/4*e^6*a^2*c/(a^3*c^3*e^6)^(1/
2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c
^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))+1/2*e^4*a*c^2/(a^3*c^3*e^6)^(1/2)/((a*c^
2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c^2*e^2*d+
(a^3*c^3*e^6)^(1/2))*a)^(1/2))*d^2+1/4*e^2/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)
^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))*
d-1/4*e^6*a^2*c/(a^3*c^3*e^6)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)
*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))+1/2*e^
4*a*c^2/(a^3*c^3*e^6)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)*arctan(
a*c*e*(e*x+d)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))*d^2-1/4*e^2/((
-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c^2*e
^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))*d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230541, size = 917, normalized size = 4.39 \[ -\frac{{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) +{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + 4 \,{\left (c d x + a e\right )} \sqrt{e x + d}}{8 \,{\left (a c^{2} x^{2} - a^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*((a*c^2*x^2 - a^2*c)*sqrt((a^3*c^2*sqrt(e^6/(a^3*c^5)) + 4*c*d^3 - 3*a*d*e^
2)/(a^3*c^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) + (2*a^3*c^4*d*sqrt(e^6/(
a^3*c^5)) + a^2*c*e^4)*sqrt((a^3*c^2*sqrt(e^6/(a^3*c^5)) + 4*c*d^3 - 3*a*d*e^2)/
(a^3*c^2))) - (a*c^2*x^2 - a^2*c)*sqrt((a^3*c^2*sqrt(e^6/(a^3*c^5)) + 4*c*d^3 -
3*a*d*e^2)/(a^3*c^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) - (2*a^3*c^4*d*sq
rt(e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt((a^3*c^2*sqrt(e^6/(a^3*c^5)) + 4*c*d^3 - 3*a
*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 - a^2*c)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3*c^5)) - 4
*c*d^3 + 3*a*d*e^2)/(a^3*c^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) + (2*a^3
*c^4*d*sqrt(e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3*c^5)) - 4*c
*d^3 + 3*a*d*e^2)/(a^3*c^2))) + (a*c^2*x^2 - a^2*c)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3
*c^5)) - 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d
) - (2*a^3*c^4*d*sqrt(e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(e^6/(a^3*c
^5)) - 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))) + 4*(c*d*x + a*e)*sqrt(e*x + d))/(a*c^2*
x^2 - a^2*c)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError