3.613 \(\int \frac{(d+e x)^{7/2}}{\left (a-c x^2\right )^2} \, dx\)
Optimal. Leaf size=263 \[ -\frac{\left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac{\left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac{e \sqrt{d+e x} \left (5 a e^2+c d^2\right )}{2 a c^2}+\frac{(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac{d e (d+e x)^{3/2}}{2 a c} \]
[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)) - ((Sqrt[c]*d - Sqrt[a]*e)^(
5/2)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
- Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)) + ((2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + S
qrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4
*a^(3/2)*c^(9/4))
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Rubi [A] time = 1.33045, antiderivative size = 263, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ -\frac{\left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac{\left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac{e \sqrt{d+e x} \left (5 a e^2+c d^2\right )}{2 a c^2}+\frac{(d+e x)^{5/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac{d e (d+e x)^{3/2}}{2 a c} \]
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)) - ((Sqrt[c]*d - Sqrt[a]*e)^(
5/2)*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
- Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)) + ((2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + S
qrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4
*a^(3/2)*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)**2,x)
[Out]
Timed out
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Mathematica [A] time = 0.686222, size = 265, normalized size = 1.01 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (5 a^2 e^3+a c e \left (3 d^2+3 d e x-4 e^2 x^2\right )+c^2 d^3 x\right )}{a-c x^2}-\frac{\left (5 \sqrt{a} e+2 \sqrt{c} d\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (2 \sqrt{c} d-5 \sqrt{a} e\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}}{4 a^{3/2} c^2} \]
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 - Sqrt[a]*e)^3*(2*Sqrt[c]*d + 5*Sqrt[a]*e)*Ar
cTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]
*Sqrt[c]*e] + ((2*Sqrt[c]*d - 5*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^3*ArcTanh[(Sq
rt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*
e])/(4*a^(3/2)*c^2)
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Maple [B] time = 0.108, size = 899, normalized size = 3.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(7/2)/(-c*x^2+a)^2,x)
[Out]
2*e^3/c^2*(e*x+d)^(1/2)-3/2*e^3/c/(c*e^2*x^2-a*e^2)*d*(e*x+d)^(3/2)-1/2*e/(c*e^2
*x^2-a*e^2)*d^3/a*(e*x+d)^(3/2)-1/2*e^5/c^2/(c*e^2*x^2-a*e^2)*a*(e*x+d)^(1/2)+1/
2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*d^4-5/4*e^8/c*a^3/(a^3*c*e^6)^(1/2)/((a*d*
e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*d*e^2*c+(a^3
*c*e^6)^(1/2))*a*c)^(1/2))-9/4*e^6*a^2/(a^3*c*e^6)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)
^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*
c)^(1/2))*d^2+1/2*e^4*c*a/(a^3*c*e^6)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^
(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d^4
-13/4*e^4/c*a/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1
/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d+1/4*e^2/((a*d*e^2*c+(a^3*c*e^6)
^(1/2))*a*c)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*
c)^(1/2))*d^3-5/4*e^8/c*a^3/(a^3*c*e^6)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*
c)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))-
9/4*e^6*a^2/(a^3*c*e^6)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctan(
a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d^2+1/2*e^4*c*a/
(a^3*c*e^6)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctan(a*c*e*(e*x+d
)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d^4+13/4*e^4/c*a/((-a*d*e^2*
c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e
^6)^(1/2))*a*c)^(1/2))*d-1/4*e^2/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2)*arct
an(a*c*e*(e*x+d)^(1/2)/((-a*d*e^2*c+(a^3*c*e^6)^(1/2))*a*c)^(1/2))*d^3
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \]
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 = 0.428617, size = 2799, normalized size = 10.64 \[ \text{result too large to display} \]
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 + 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))*lo
g((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 + 625*a^4*e^1
4)/(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 + 7
700*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 - 17
71*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 + 21
966*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