3.611 \(\int \frac{1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=663 \[ -\frac{2 e}{\sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

[Out]

(-2*e)/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (c^(1/4)*e*(2*Sqrt[c]*d - Sqrt[c*d^2 +
a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*S
qrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(2*Sqrt[c]*d - Sqrt[c*d^2 + a
*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d +
e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sq
rt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(2*Sqrt[c]*d + Sqrt[c*d^2 + a*
e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*
e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sqrt[
Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*e*(2*Sqrt[c]*d + Sqrt[c*d^2 + a*e^2
])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2
]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqr
t[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 2.51002, antiderivative size = 663, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{2 e}{\sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (\sqrt{a e^2+c d^2}+2 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt [4]{c} e \left (2 \sqrt{c} d-\sqrt{a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a + c*x^2)),x]

[Out]

(-2*e)/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (c^(1/4)*e*(2*Sqrt[c]*d - Sqrt[c*d^2 +
a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*S
qrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(2*Sqrt[c]*d - Sqrt[c*d^2 + a
*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d +
e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sq
rt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(2*Sqrt[c]*d + Sqrt[c*d^2 + a*
e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*
e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sqrt[
Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*e*(2*Sqrt[c]*d + Sqrt[c*d^2 + a*e^2
])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2
]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqr
t[c]*d + Sqrt[c*d^2 + a*e^2]])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+a),x)

[Out]

Timed out

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Mathematica [C]  time = 0.806863, size = 209, normalized size = 0.32 \[ \frac{\frac{\sqrt{c} \left (\sqrt{a} e-i \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{\sqrt{c} \left (\sqrt{a} e+i \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \sqrt{c d+i \sqrt{a} \sqrt{c} e}}-\frac{2 e}{\sqrt{d+e x}}}{a e^2+c d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)),x]

[Out]

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

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Maple [B]  time = 0.072, size = 5655, normalized size = 8.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+a),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)*(e*x + d)^(3/2)), x)

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Fricas [A]  time = 0.251657, size = 3802, normalized size = 5.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((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(-(c^2*d^3 - 3*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(-(9*c^3*d^4*e^2 - 6*a*c^2*d^
2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^
4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*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(-(3*c^2*d^2*e - a*c*e^3)*sq
rt(e*x + d) + (6*a*c^2*d^3*e^2 - 2*a^2*c*d*e^4 + (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2
- 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)
/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15
*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*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(-(9*c^3*d^4*e^
2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d
^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*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(-(c^2*d^3 - 3*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(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(
a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a
^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*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(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) - (6*a*c
^2*d^3*e^2 - 2*a^2*c*d*e^4 + (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 -
a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2
*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6
*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*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(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4
+ a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*
d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*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
(-(c^2*d^3 - 3*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(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c
^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a
^6*c*d^2*e^10 + a^7*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(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 - 2*a^2*c
*d*e^4 - (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^
3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a
^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^
7*e^12)))*sqrt(-(c^2*d^3 - 3*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(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*
d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2
*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*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(-(c^2*d^3 - 3*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(-(9*c^3*
d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3
*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*
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(-(3*c^2
*d^2*e - a*c*e^3)*sqrt(e*x + d) - (6*a*c^2*d^3*e^2 - 2*a^2*c*d*e^4 - (a*c^4*d^8
+ 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*
d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*
a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2
*d^3 - 3*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(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^1
0*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d
^2*e^10 + a^7*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*e)/((c*d^2 + a*e^2)*sqrt(e*x + d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+a),x)

[Out]

Integral(1/((a + c*x**2)*(d + e*x)**(3/2)), 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(1/((c*x^2 + a)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out