3.1997 \(\int \frac{1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx\)

Optimal. Leaf size=186 \[ -\frac{2 c^{7/2} d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}+\frac{2 c^3 d^3}{\sqrt{d+e x} \left (c d^2-a e^2\right )^4}+\frac{2 c^2 d^2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{2 c d}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac{2}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^(7/2)) + (2*c*d)/(5*(c*d^2 - a*e^2)^2*(d + e*x)^(
5/2)) + (2*c^2*d^2)/(3*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)) + (2*c^3*d^3)/((c*d^2
- a*e^2)^4*Sqrt[d + e*x]) - (2*c^(7/2)*d^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d +
e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)

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Rubi [A]  time = 0.453451, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 c^{7/2} d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}+\frac{2 c^3 d^3}{\sqrt{d+e x} \left (c d^2-a e^2\right )^4}+\frac{2 c^2 d^2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{2 c d}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac{2}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^(7/2)) + (2*c*d)/(5*(c*d^2 - a*e^2)^2*(d + e*x)^(
5/2)) + (2*c^2*d^2)/(3*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)) + (2*c^3*d^3)/((c*d^2
- a*e^2)^4*Sqrt[d + e*x]) - (2*c^(7/2)*d^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d +
e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)

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Rubi in Sympy [A]  time = 79.0861, size = 167, normalized size = 0.9 \[ \frac{2 c^{\frac{7}{2}} d^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{9}{2}}} + \frac{2 c^{3} d^{3}}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{4}} - \frac{2 c^{2} d^{2}}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{2 c d}{5 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2}{7 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

2*c**(7/2)*d**(7/2)*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2))/(a
*e**2 - c*d**2)**(9/2) + 2*c**3*d**3/(sqrt(d + e*x)*(a*e**2 - c*d**2)**4) - 2*c*
*2*d**2/(3*(d + e*x)**(3/2)*(a*e**2 - c*d**2)**3) + 2*c*d/(5*(d + e*x)**(5/2)*(a
*e**2 - c*d**2)**2) - 2/(7*(d + e*x)**(7/2)*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.803142, size = 178, normalized size = 0.96 \[ \frac{2 \left (\frac{15 c^3 d^3 (d+e x)^3}{\left (c d^2-a e^2\right )^4}+\frac{5 c^2 d^2 (d+e x)^2}{\left (c d^2-a e^2\right )^3}+\frac{3 c d (d+e x)}{\left (c d^2-a e^2\right )^2}+\frac{15}{7 c d^2-7 a e^2}\right )}{15 (d+e x)^{7/2}}-\frac{2 c^{7/2} d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

(2*(15/(7*c*d^2 - 7*a*e^2) + (3*c*d*(d + e*x))/(c*d^2 - a*e^2)^2 + (5*c^2*d^2*(d
+ e*x)^2)/(c*d^2 - a*e^2)^3 + (15*c^3*d^3*(d + e*x)^3)/(c*d^2 - a*e^2)^4))/(15*
(d + e*x)^(7/2)) - (2*c^(7/2)*d^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sq
rt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)

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Maple [A]  time = 0.019, size = 175, normalized size = 0.9 \[ -{\frac{2}{7\,a{e}^{2}-7\,c{d}^{2}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}}-{\frac{2\,{c}^{2}{d}^{2}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,cd}{5\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}+2\,{\frac{{c}^{3}{d}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}\sqrt{ex+d}}}+2\,{\frac{{c}^{4}{d}^{4}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(7/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

-2/7/(a*e^2-c*d^2)/(e*x+d)^(7/2)-2/3*c^2*d^2/(a*e^2-c*d^2)^3/(e*x+d)^(3/2)+2/5*c
*d/(a*e^2-c*d^2)^2/(e*x+d)^(5/2)+2*c^3*d^3/(a*e^2-c*d^2)^4/(e*x+d)^(1/2)+2*c^4*d
^4/(a*e^2-c*d^2)^4/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*
d^2)*c*d)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232426, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/105*(210*c^3*d^3*e^3*x^3 + 352*c^3*d^6 - 244*a*c^2*d^4*e^2 + 132*a^2*c*d^2*e^
4 - 30*a^3*e^6 + 70*(10*c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 105*(c^3*d^3*e^3*x^3
+ 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^6)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 - a
*e^2))*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c*d
/(c*d^2 - a*e^2)))/(c*d*x + a*e)) + 14*(58*c^3*d^5*e - 16*a*c^2*d^3*e^3 + 3*a^2*
c*d*e^5)*x)/((c^4*d^11 - 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 - 4*a^3*c*d^5*e^6 +
a^4*d^3*e^8 + (c^4*d^8*e^3 - 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 - 4*a^3*c*d^2*
e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 - 4*a*c^3*d^7*e^4 + 6*a^2*c^2*d^5*e^6 - 4*a
^3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e - 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6
*e^5 - 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x)*sqrt(e*x + d)), 2/105*(105*c^3*d^3*e^3*
x^3 + 176*c^3*d^6 - 122*a*c^2*d^4*e^2 + 66*a^2*c*d^2*e^4 - 15*a^3*e^6 + 35*(10*c
^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 - 105*(c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c
^3*d^5*e*x + c^3*d^6)*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 -
a*e^2)*sqrt(-c*d/(c*d^2 - a*e^2))/(sqrt(e*x + d)*c*d)) + 7*(58*c^3*d^5*e - 16*a*
c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/((c^4*d^11 - 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4
- 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4*d^8*e^3 - 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^
4*e^7 - 4*a^3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 - 4*a*c^3*d^7*e^4 + 6*a
^2*c^2*d^5*e^6 - 4*a^3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e - 4*a*c^3*d^8
*e^3 + 6*a^2*c^2*d^6*e^5 - 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x)*sqrt(e*x + d))]

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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(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

Timed out