3.621 \(\int \frac{(d+e x)^{5/2}}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=811 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac{d e \sqrt{d+e x}}{2 a c}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

[Out]

-(d*e*Sqrt[d + e*x])/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a + c*x^2
)) + (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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]]])/(4*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]
*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 + Sq
rt[c*d^2 + a*e^2]*(c*d^2 + 3*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]]])/(4*
Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (
e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*a*e^2))*Log[Sq
rt[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)])/(8*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt
[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 +
a*e^2]*(c*d^2 + 3*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)])/(8*Sqrt[2]*a*c^(7
/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

_______________________________________________________________________________________

Rubi [A]  time = 6.223, antiderivative size = 811, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{2 a c \left (c x^2+a\right )}-\frac{d e \sqrt{d+e x}}{2 a c}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^{3/2} d^3+a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (c d^2+3 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*e*Sqrt[d + e*x])/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a + c*x^2
)) + (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*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]]])/(4*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]
*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 + Sq
rt[c*d^2 + a*e^2]*(c*d^2 + 3*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]]])/(4*
Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (
e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(c*d^2 + 3*a*e^2))*Log[Sq
rt[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)])/(8*Sqrt[2]*a*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt
[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c^(3/2)*d^3 + a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 +
a*e^2]*(c*d^2 + 3*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)])/(8*Sqrt[2]*a*c^(7
/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.456171, size = 271, normalized size = 0.33 \[ \frac{\frac{2 \sqrt{a} c \sqrt{d+e x} \left (c d^2 x-a e (2 d+e x)\right )}{a+c x^2}+\frac{\sqrt{c} \left (3 \sqrt{a} e-2 i \sqrt{c} d\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{\sqrt{c} \left (\sqrt{c} d+i \sqrt{a} e\right )^2 \left (3 \sqrt{a} e+2 i \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{4 a^{3/2} c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.217, size = 6355, normalized size = 7.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.26698, size = 1867, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*
c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))
*log((20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(
e*x + d) + (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2
)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2
*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^
2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) - (a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*
d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2
*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d^4*e
^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(e*x + d) - (5*a^2*c^3*d^3*e^4 + 9*a^3*
c^2*d*e^6 - (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e
^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 +
a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*
c^3))) + (a*c^2*x^2 + a^2*c)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 -
a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c
^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*s
qrt(e*x + d) + (5*a^2*c^3*d^3*e^4 + 9*a^3*c^2*d*e^6 + (2*a^3*c^6*d^2 + 3*a^4*c^5
*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4
*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*
c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) - (a*c^2*x^2 + a^2*c)*sqrt(-(4*
c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c
*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log((20*c^3*d^6*e^3 + 101*a*c^2*d
^4*e^5 + 162*a^2*c*d^2*e^7 + 81*a^3*e^9)*sqrt(e*x + d) - (5*a^2*c^3*d^3*e^4 + 9*
a^3*c^2*d*e^6 + (2*a^3*c^6*d^2 + 3*a^4*c^5*e^2)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d
^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e
^4 - a^3*c^3*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(
a^3*c^3))) - 4*(2*a*d*e - (c*d^2 - a*e^2)*x)*sqrt(e*x + d))/(a*c^2*x^2 + a^2*c)

_______________________________________________________________________________________

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)**(5/2)/(c*x**2+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError