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

Optimal. Leaf size=887 \[ -\frac{(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}-\frac{d e (d+e x)^{3/2}}{2 a c}-\frac{e \left (c d^2-5 a e^2\right ) \sqrt{d+e x}}{2 a c^2}+\frac{e \left (c^2 d^4-4 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^2 d^4-4 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^2 d^4-4 a c e^2 d^2-\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^2 d^4-4 a c e^2 d^2-\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

[Out]

-(e*(c*d^2 - 5*a*e^2)*Sqrt[d + e*x])/(2*a*c^2) - (d*e*(d + e*x)^(3/2))/(2*a*c) -
((a*e - c*d*x)*(d + e*x)^(5/2))/(2*a*c*(a + c*x^2)) + (e*(c^2*d^4 - 4*a*c*d^2*e
^2 - 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]
*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 + Sqrt[c]*d
*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])
- (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^
2 + 13*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^(9/4)*Sqrt[c*d
^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c^2*d^4 - 4*a*c*d^2*e^2
- 5*a^2*e^4 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + S
qrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 14.2013, antiderivative size = 887, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}-\frac{d e (d+e x)^{3/2}}{2 a c}-\frac{e \left (c d^2-5 a e^2\right ) \sqrt{d+e x}}{2 a c^2}+\frac{e \left (c^2 d^4-4 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^2 d^4-4 a c e^2 d^2+\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{4 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c^2 d^4-4 a c e^2 d^2-\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c^2 d^4-4 a c e^2 d^2-\sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{8 \sqrt{2} a c^{9/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

-(e*(c*d^2 - 5*a*e^2)*Sqrt[d + e*x])/(2*a*c^2) - (d*e*(d + e*x)^(3/2))/(2*a*c) -
((a*e - c*d*x)*(d + e*x)^(5/2))/(2*a*c*(a + c*x^2)) + (e*(c^2*d^4 - 4*a*c*d^2*e
^2 - 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]
*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 + Sqrt[c]*d
*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])
- (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^
2 + 13*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^(9/4)*Sqrt[c*d
^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c^2*d^4 - 4*a*c*d^2*e^2
- 5*a^2*e^4 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + S
qrt[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)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 0.513956, size = 282, normalized size = 0.32 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (5 a^2 e^3+a c e \left (-3 d^2-3 d e x+4 e^2 x^2\right )+c^2 d^3 x\right )}{a+c x^2}+\frac{\left (5 \sqrt{a} e-2 i \sqrt{c} d\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{c} d+i \sqrt{a} e\right )^3 \left (5 \sqrt{a} e+2 i \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{4 a^{3/2} c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.216, size = 8005, normalized size = 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)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.41985, size = 2823, normalized size = 3.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*((a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e
^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 219
66*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))
*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c
^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6
*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*
d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2
*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7
+ 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4
*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 +
625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 +
35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d
^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 62
5*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 +
6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^1
3)*sqrt(e*x + d) - (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^
8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6
+ 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*
e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a
^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2
*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) + (a*c^3
*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3
*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d
^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(140*c
^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 +
2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 - 21*a^
3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + 2*(a^3*c^8*d^3 + 4*a^4*
c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^1
0 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*
d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 1
0780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14
)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^
5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 107
80*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/
(a^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^
3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*x
+ d) - (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*
c^2*e^10 + 2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c
^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c
^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a
^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 -
7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - 4*(4*a*c*e^3*x^2 -
3*a*c*d^2*e + 5*a^2*e^3 + (c^2*d^3 - 3*a*c*d*e^2)*x)*sqrt(e*x + d))/(a*c^3*x^2
+ a^2*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)**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)^(7/2)/(c*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError