3.633 \(\int \frac{(d+e x)^{5/2}}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=846 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (c x^2+a\right )^2}-\frac{3 \left (a d e-\left (2 c d^2+a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c \left (c x^2+a\right )}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

[Out]

-((a*e - c*d*x)*(d + e*x)^(3/2))/(4*a*c*(a + c*x^2)^2) - (3*Sqrt[d + e*x]*(a*d*e
- (2*c*d^2 + a*e^2)*x))/(16*a^2*c*(a + c*x^2)) + (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt
[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*c*d^2 + a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqr
t[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]]])/(32*Sqrt[2]*a^2*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2
*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]]])/(32*Sqrt[2]*a^2*c^(7/4)
*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^
3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*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] + S
qrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d +
Sqrt[c*d^2 + a*e^2]]) + (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a
*e^2]*(2*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)])/(64*Sqrt[2]*a^2*c^(
7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 7.04692, antiderivative size = 846, 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 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (c x^2+a\right )^2}-\frac{3 \left (a d e-\left (2 c d^2+a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c \left (c x^2+a\right )}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(a + c*x^2)^3,x]

[Out]

-((a*e - c*d*x)*(d + e*x)^(3/2))/(4*a*c*(a + c*x^2)^2) - (3*Sqrt[d + e*x]*(a*d*e
- (2*c*d^2 + a*e^2)*x))/(16*a^2*c*(a + c*x^2)) + (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt
[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*c*d^2 + a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqr
t[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]]])/(32*Sqrt[2]*a^2*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2
*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]]])/(32*Sqrt[2]*a^2*c^(7/4)
*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^
3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*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] + S
qrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d +
Sqrt[c*d^2 + a*e^2]]) + (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a
*e^2]*(2*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)])/(64*Sqrt[2]*a^2*c^(
7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[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((e*x+d)**(5/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.729556, size = 311, normalized size = 0.37 \[ \frac{-\frac{3 \left (a^{3/2} \sqrt{c} e^3+2 \sqrt{a} c^{3/2} d^2 e+3 i a c d e^2+4 i c^2 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{2 \sqrt{a} c \sqrt{d+e x} \left (-a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )+6 c^2 d^2 x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt{c d+i \sqrt{a} \sqrt{c} e} \left (2 \sqrt{a} \sqrt{c} d e+i a e^2+4 i c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{c d+i \sqrt{a} \sqrt{c} e}}{\sqrt{c} d+i \sqrt{a} e}\right )}{32 a^{5/2} c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(a + c*x^2)^3,x]

[Out]

((2*Sqrt[a]*c*Sqrt[d + e*x]*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) + a*c*x*(10*d^2 +
d*e*x + 3*e^2*x^2)))/(a + c*x^2)^2 - (3*((4*I)*c^2*d^3 + 2*Sqrt[a]*c^(3/2)*d^2*
e + (3*I)*a*c*d*e^2 + a^(3/2)*Sqrt[c]*e^3)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[
c*d - I*Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e] + 3*Sqrt[c*d + I*Sq
rt[a]*Sqrt[c]*e]*((4*I)*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e + I*a*e^2)*ArcTanh[(Sqrt[c
*d + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(32*a^(5/2)
*c^2)

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Maple [B]  time = 0.217, size = 6426, normalized size = 7.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(c*x^2+a)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(5/2)/(c*x^2 + a)^3, x)

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Fricas [A]  time = 0.266648, size = 1400, normalized size = 1.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/64*(3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(-e^10/(a^5*c^7
)) + 16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^
5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d*e^6 - (4*a^5*c^6*d
^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(-e^10/(a^5*c^7)) +
16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) - 3*(a^2*c^3*x^4 + 2*a^3*
c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*
e^2 + 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)
*sqrt(e*x + d) - 27*(2*a^3*c^2*d*e^6 - (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/
(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*e^2 +
5*a^2*d*e^4)/(a^5*c^3))) + 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^5*c^
3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*
log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2
*d*e^6 + (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt
(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))) - 3*(
a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^
2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 + 12*a*c
*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*a^3*c^2*d*e^6 + (4*a^5*c^6*d^2 + a^6*c
^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5
- 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))) + 4*(a*c*d*e*x^2 - 7*a^2*d*e + 3*(2*
c^2*d^2 + a*c*e^2)*x^3 + (10*a*c*d^2 - a^2*e^2)*x)*sqrt(e*x + d))/(a^2*c^3*x^4 +
2*a^3*c^2*x^2 + a^4*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)**(5/2)/(c*x**2+a)**3,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)^(5/2)/(c*x^2 + a)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError