3.632 \(\int \frac{(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=905 \[ -\frac{(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}-\frac{\left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c^2 \left (c x^2+a\right )}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

[Out]

-((a*e - c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a + c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(7*
c*d^2 + 5*a*e^2) - 2*c*d*(3*c*d^2 + 2*a*e^2)*x))/(16*a^2*c^2*(a + c*x^2)) + (e*(
6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(6*c*d^2
+ 8*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(9/4)*Sqrt
[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^2*d^4 + 11*a*c*
d^2*e^2 + 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 8*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[Sq
rt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqr
t[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4
- 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(3*c*d^2 + 4*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*
d^2 + a*e^2]]) + (e*(6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c
*d^2 + a*e^2]*(3*c*d^2 + 4*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqr
t[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^(9/4)*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 = 12.6937, antiderivative size = 905, 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 \[ -\frac{(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}-\frac{\left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c^2 \left (c x^2+a\right )}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (6 c d^2+8 a e^2\right ) d+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^2 d^4+11 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (3 c d^2+4 a e^2\right ) d+5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a + c*x^2)^3,x]

[Out]

-((a*e - c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a + c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(7*
c*d^2 + 5*a*e^2) - 2*c*d*(3*c*d^2 + 2*a*e^2)*x))/(16*a^2*c^2*(a + c*x^2)) + (e*(
6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(6*c*d^2
+ 8*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(9/4)*Sqrt
[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^2*d^4 + 11*a*c*
d^2*e^2 + 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 8*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[Sq
rt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqr
t[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4
- 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(3*c*d^2 + 4*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*
d^2 + a*e^2]]) + (e*(6*c^2*d^4 + 11*a*c*d^2*e^2 + 5*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c
*d^2 + a*e^2]*(3*c*d^2 + 4*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqr
t[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^(9/4)*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)**(7/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.676836, size = 337, normalized size = 0.37 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (-5 a^3 e^3-a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )+6 c^3 d^3 x^3\right )}{\left (a+c x^2\right )^2}+\frac{\left (18 \sqrt{a} \sqrt{c} d e+5 i a e^2-12 i c d^2\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{c} d+i \sqrt{a} e\right )^2 \left (18 \sqrt{a} \sqrt{c} d e-5 i a e^2+12 i c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{32 a^{5/2} c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a + c*x^2)^3,x]

[Out]

((2*Sqrt[a]*Sqrt[d + e*x]*(-5*a^3*e^3 + 6*c^3*d^3*x^3 + a*c^2*d*x*(10*d^2 + d*e*
x + 8*e^2*x^2) - a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)))/(a + c*x^2)^2 + ((Sqrt
[c]*d - I*Sqrt[a]*e)^2*((-12*I)*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + (5*I)*a*e^2)*Ar
cTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - I*Sqr
t[a]*Sqrt[c]*e] + ((Sqrt[c]*d + I*Sqrt[a]*e)^2*((12*I)*c*d^2 + 18*Sqrt[a]*Sqrt[c
]*d*e - (5*I)*a*e^2)*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])/(32*a^(5/2)*c^2)

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Maple [B]  time = 0.247, size = 9831, normalized size = 10.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(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. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + a)^3, x)

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Fricas [A]  time = 0.326888, size = 2364, normalized size = 2.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/64*((a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5
*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 105
0*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((3024*c^4*d^8*e^5 + 10
908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*
sqrt(e*x + d) + (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 +
(12*a^5*c^8*d^3 + 13*a^6*c^7*d*e^2)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12
+ 625*a^2*e^14)/(a^5*c^9)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d
^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 6
25*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))) - (a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*s
qrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*
c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5
*c^4))*log((3024*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 625
0*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 + 255*a^4*
c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 + 13*a^6*c^7*d*e^2)*sqrt(-(441*
c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt(-(144*c^3*d^7
+ 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2
*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))) + (a^2*c^4
*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2
*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12
+ 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((3024*c^4*d^8*e^5 + 10908*a*c^3*d^6*
e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d)
+ (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^
3 + 13*a^6*c^7*d*e^2)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14
)/(a^5*c^9)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a
^3*d*e^6 - a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(
a^5*c^9)))/(a^5*c^4))) - (a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*
d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*sqrt(-(441
*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((30
24*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^3*c*d^2*e^
11 + 625*a^4*e^13)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3*d^2*e^8 +
125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 + 13*a^6*c^7*d*e^2)*sqrt(-(441*c^2*d^4*e^10 +
1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^
5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 10
50*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))) - 4*(11*a^2*c*d^2*e + 5*
a^3*e^3 - 2*(3*c^3*d^3 + 4*a*c^2*d*e^2)*x^3 - (a*c^2*d^2*e - 9*a^2*c*e^3)*x^2 -
2*(5*a*c^2*d^3 - 2*a^2*c*d*e^2)*x)*sqrt(e*x + d))/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 +
a^4*c^2)

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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)**(7/2)/(c*x**2+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 89.9026, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + a)^3,x, algorithm="giac")

[Out]

Done