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

Optimal. Leaf size=294 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac{\sqrt{d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

[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)) - ((Sqr
t[c]*d - Sqrt[a]*e)^(3/2)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[
(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4)) + ((S
qrt[c]*d + Sqrt[a]*e)^(3/2)*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTan
h[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

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Rubi [A]  time = 1.24375, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac{\sqrt{d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{(d+e x)^{5/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

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)) - ((Sqr
t[c]*d - Sqrt[a]*e)^(3/2)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[
(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4)) + ((S
qrt[c]*d + Sqrt[a]*e)^(3/2)*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTan
h[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

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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 [A]  time = 0.894803, size = 315, normalized size = 1.07 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (-5 a^3 e^3+a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )-6 c^3 d^3 x^3\right )}{\left (a-c x^2\right )^2}-\frac{\left (18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^2 \left (-18 \sqrt{a} \sqrt{c} d e+5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}}{32 a^{5/2} c^2} \]

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 - Sqrt[a]*e)^2*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(Sqrt
[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]
+ ((Sqrt[c]*d + Sqrt[a]*e)^2*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt[a]*S
qrt[c]*e])/(32*a^(5/2)*c^2)

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Maple [B]  time = 0.118, size = 1320, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e^3/(c*e^2*x^2-a*e^2)^2*d/a*(e*x+d)^(7/2)-3/8*e/(c*e^2*x^2-a*e^2)^2*d^3/a^2*
(e*x+d)^(7/2)*c+9/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(5/2)-23/16*e^3/(c*e^2*x^
2-a*e^2)^2/a*(e*x+d)^(5/2)*d^2+9/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(5/2)*d^4
-7/8*e^5/(c*e^2*x^2-a*e^2)^2*d/c*(e*x+d)^(3/2)+2*e^3/(c*e^2*x^2-a*e^2)^2*d^3/a*(
e*x+d)^(3/2)-9/8*e/(c*e^2*x^2-a*e^2)^2*d^5/a^2*c*(e*x+d)^(3/2)-5/16*e^7/(c*e^2*x
^2-a*e^2)^2*a/c^2*(e*x+d)^(1/2)+e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*d^2-17/1
6*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*d^4+3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e
*x+d)^(1/2)*d^6+5/32*e^11*a^3*c^2/(a^5*c^5*e^10)^(1/2)/((e^4*a^2*c^3*d+(a^5*c^5*
e^10)^(1/2))*c)^(1/2)*arctanh(e^2*a*c^2*(e*x+d)^(1/2)/((e^4*a^2*c^3*d+(a^5*c^5*e
^10)^(1/2))*c)^(1/2))-19/32*e^9*a^2*c^3/(a^5*c^5*e^10)^(1/2)/((e^4*a^2*c^3*d+(a^
5*c^5*e^10)^(1/2))*c)^(1/2)*arctanh(e^2*a*c^2*(e*x+d)^(1/2)/((e^4*a^2*c^3*d+(a^5
*c^5*e^10)^(1/2))*c)^(1/2))*d^2+3/8*e^7*a*c^4/(a^5*c^5*e^10)^(1/2)/((e^4*a^2*c^3
*d+(a^5*c^5*e^10)^(1/2))*c)^(1/2)*arctanh(e^2*a*c^2*(e*x+d)^(1/2)/((e^4*a^2*c^3*
d+(a^5*c^5*e^10)^(1/2))*c)^(1/2))*d^4-1/4*e^5/((e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/
2))*c)^(1/2)*arctanh(e^2*a*c^2*(e*x+d)^(1/2)/((e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2
))*c)^(1/2))*d+3/16*e^3/a*c/((e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^(1/2)*arcta
nh(e^2*a*c^2*(e*x+d)^(1/2)/((e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^(1/2))*d^3+5
/32*e^11*a^3*c^2/(a^5*c^5*e^10)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^
(1/2)*arctan(e^2*a*c^2*(e*x+d)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^(
1/2))-19/32*e^9*a^2*c^3/(a^5*c^5*e^10)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/
2))*c)^(1/2)*arctan(e^2*a*c^2*(e*x+d)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2
))*c)^(1/2))*d^2+3/8*e^7*a*c^4/(a^5*c^5*e^10)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^
10)^(1/2))*c)^(1/2)*arctan(e^2*a*c^2*(e*x+d)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^1
0)^(1/2))*c)^(1/2))*d^4+1/4*e^5/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^(1/2)*
arctan(e^2*a*c^2*(e*x+d)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^(1/2))*
d-3/16*e^3/a*c/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^(1/2)*arctan(e^2*a*c^2*
(e*x+d)^(1/2)/((-e^4*a^2*c^3*d+(a^5*c^5*e^10)^(1/2))*c)^(1/2))*d^3

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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 = 0.323818, size = 2336, normalized size = 7.95 \[ \text{result too large to display} \]

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 - 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 - 1090
8*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)*sq
rt(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 - (1
2*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 + 6
25*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*sq
rt((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((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 - 1
050*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 + 62
5*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

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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(-(e*x + d)^(7/2)/(c*x^2 - a)^3,x, algorithm="giac")

[Out]

Timed out