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

Optimal. Leaf size=187 \[ -\frac{2 e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{7/2}}-\frac{2 e (2 c d-b e)}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 e}{5 d (d+e x)^{5/2} (c d-b e)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{7/2}} \]

[Out]

(-2*e)/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (2*e*(2*c*d - b*e))/(3*d^2*(c*d - b*e
)^2*(d + e*x)^(3/2)) - (2*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2))/(d^3*(c*d - b*e)^
3*Sqrt[d + e*x]) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(7/2)) + (2*c^(7/2)*A
rcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(7/2))

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Rubi [A]  time = 0.832185, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{2 e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{7/2}}-\frac{2 e (2 c d-b e)}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 e}{5 d (d+e x)^{5/2} (c d-b e)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e)/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (2*e*(2*c*d - b*e))/(3*d^2*(c*d - b*e
)^2*(d + e*x)^(3/2)) - (2*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2))/(d^3*(c*d - b*e)^
3*Sqrt[d + e*x]) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(7/2)) + (2*c^(7/2)*A
rcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(7/2))

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Rubi in Sympy [A]  time = 100.344, size = 170, normalized size = 0.91 \[ \frac{2 e}{5 d \left (d + e x\right )^{\frac{5}{2}} \left (b e - c d\right )} + \frac{2 e \left (b e - 2 c d\right )}{3 d^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )^{2}} + \frac{2 e \left (b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right )}{d^{3} \sqrt{d + e x} \left (b e - c d\right )^{3}} + \frac{2 c^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \left (b e - c d\right )^{\frac{7}{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e/(5*d*(d + e*x)**(5/2)*(b*e - c*d)) + 2*e*(b*e - 2*c*d)/(3*d**2*(d + e*x)**(3
/2)*(b*e - c*d)**2) + 2*e*(b**2*e**2 - 3*b*c*d*e + 3*c**2*d**2)/(d**3*sqrt(d + e
*x)*(b*e - c*d)**3) + 2*c**(7/2)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b*
(b*e - c*d)**(7/2)) - 2*atanh(sqrt(d + e*x)/sqrt(d))/(b*d**(7/2))

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Mathematica [A]  time = 1.26854, size = 182, normalized size = 0.97 \[ -\frac{2 e \left (b^2 e^2 \left (23 d^2+35 d e x+15 e^2 x^2\right )-3 b c d e \left (22 d^2+35 d e x+15 e^2 x^2\right )+c^2 d^2 \left (58 d^2+100 d e x+45 e^2 x^2\right )\right )}{15 d^3 (d+e x)^{5/2} (c d-b e)^3}+\frac{2 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{7/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*(-3*b*c*d*e*(22*d^2 + 35*d*e*x + 15*e^2*x^2) + b^2*e^2*(23*d^2 + 35*d*e*x
+ 15*e^2*x^2) + c^2*d^2*(58*d^2 + 100*d*e*x + 45*e^2*x^2)))/(15*d^3*(c*d - b*e)^
3*(d + e*x)^(5/2)) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(7/2)) + (2*c^(7/2)
*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(7/2))

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Maple [A]  time = 0.027, size = 228, normalized size = 1.2 \[{\frac{2\,b{e}^{2}}{3\,{d}^{2} \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,ce}{3\,d \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{{b}^{2}{e}^{3}}{{d}^{3} \left ( be-cd \right ) ^{3}\sqrt{ex+d}}}-6\,{\frac{bc{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{3}\sqrt{ex+d}}}+6\,{\frac{e{c}^{2}}{d \left ( be-cd \right ) ^{3}\sqrt{ex+d}}}+{\frac{2\,e}{5\,d \left ( be-cd \right ) } \left ( ex+d \right ) ^{-{\frac{5}{2}}}}+2\,{\frac{{c}^{4}}{ \left ( be-cd \right ) ^{3}b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{1}{b{d}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3/d^2/(b*e-c*d)^2/(e*x+d)^(3/2)*b*e^2-4/3*e/d/(b*e-c*d)^2/(e*x+d)^(3/2)*c+2/d^
3/(b*e-c*d)^3/(e*x+d)^(1/2)*b^2*e^3-6/d^2/(b*e-c*d)^3/(e*x+d)^(1/2)*b*c*e^2+6*e/
d/(b*e-c*d)^3/(e*x+d)^(1/2)*c^2+2/5*e/d/(b*e-c*d)/(e*x+d)^(5/2)+2/(b*e-c*d)^3*c^
4/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))-2*arctanh((e
*x+d)^(1/2)/d^(1/2))/b/d^(7/2)

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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*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01273, 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*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[-1/15*(15*(c^3*d^3*e^2*x^2 + 2*c^3*d^4*e*x + c^3*d^5)*sqrt(e*x + d)*sqrt(d)*sqr
t(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(
c*d - b*e)))/(c*x + b)) - 15*(c^3*d^5 - 3*b*c^2*d^4*e + 3*b^2*c*d^3*e^2 - b^3*d^
2*e^3 + (c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*c*d*e^4 - b^3*e^5)*x^2 + 2*(c^3*d
^4*e - 3*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*sqrt(e*x + d)*log(((e*x
+ 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x) + 2*(58*b*c^2*d^4*e - 66*b^2*c*d^3*e^2 +
23*b^3*d^2*e^3 + 15*(3*b*c^2*d^2*e^3 - 3*b^2*c*d*e^4 + b^3*e^5)*x^2 + 5*(20*b*c
^2*d^3*e^2 - 21*b^2*c*d^2*e^3 + 7*b^3*d*e^4)*x)*sqrt(d))/((b*c^3*d^8 - 3*b^2*c^2
*d^7*e + 3*b^3*c*d^6*e^2 - b^4*d^5*e^3 + (b*c^3*d^6*e^2 - 3*b^2*c^2*d^5*e^3 + 3*
b^3*c*d^4*e^4 - b^4*d^3*e^5)*x^2 + 2*(b*c^3*d^7*e - 3*b^2*c^2*d^6*e^2 + 3*b^3*c*
d^5*e^3 - b^4*d^4*e^4)*x)*sqrt(e*x + d)*sqrt(d)), 1/15*(30*(c^3*d^3*e^2*x^2 + 2*
c^3*d^4*e*x + c^3*d^5)*sqrt(e*x + d)*sqrt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d -
b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + 15*(c^3*d^5 - 3*b*c^2*d^4*e + 3*
b^2*c*d^3*e^2 - b^3*d^2*e^3 + (c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*c*d*e^4 - b
^3*e^5)*x^2 + 2*(c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*s
qrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x) - 2*(58*b*c^2*d^4*
e - 66*b^2*c*d^3*e^2 + 23*b^3*d^2*e^3 + 15*(3*b*c^2*d^2*e^3 - 3*b^2*c*d*e^4 + b^
3*e^5)*x^2 + 5*(20*b*c^2*d^3*e^2 - 21*b^2*c*d^2*e^3 + 7*b^3*d*e^4)*x)*sqrt(d))/(
(b*c^3*d^8 - 3*b^2*c^2*d^7*e + 3*b^3*c*d^6*e^2 - b^4*d^5*e^3 + (b*c^3*d^6*e^2 -
3*b^2*c^2*d^5*e^3 + 3*b^3*c*d^4*e^4 - b^4*d^3*e^5)*x^2 + 2*(b*c^3*d^7*e - 3*b^2*
c^2*d^6*e^2 + 3*b^3*c*d^5*e^3 - b^4*d^4*e^4)*x)*sqrt(e*x + d)*sqrt(d)), -1/15*(1
5*(c^3*d^3*e^2*x^2 + 2*c^3*d^4*e*x + c^3*d^5)*sqrt(e*x + d)*sqrt(-d)*sqrt(c/(c*d
- b*e))*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*
e)))/(c*x + b)) - 30*(c^3*d^5 - 3*b*c^2*d^4*e + 3*b^2*c*d^3*e^2 - b^3*d^2*e^3 +
(c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*c*d*e^4 - b^3*e^5)*x^2 + 2*(c^3*d^4*e - 3
*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*
x + d)*sqrt(-d))) + 2*(58*b*c^2*d^4*e - 66*b^2*c*d^3*e^2 + 23*b^3*d^2*e^3 + 15*(
3*b*c^2*d^2*e^3 - 3*b^2*c*d*e^4 + b^3*e^5)*x^2 + 5*(20*b*c^2*d^3*e^2 - 21*b^2*c*
d^2*e^3 + 7*b^3*d*e^4)*x)*sqrt(-d))/((b*c^3*d^8 - 3*b^2*c^2*d^7*e + 3*b^3*c*d^6*
e^2 - b^4*d^5*e^3 + (b*c^3*d^6*e^2 - 3*b^2*c^2*d^5*e^3 + 3*b^3*c*d^4*e^4 - b^4*d
^3*e^5)*x^2 + 2*(b*c^3*d^7*e - 3*b^2*c^2*d^6*e^2 + 3*b^3*c*d^5*e^3 - b^4*d^4*e^4
)*x)*sqrt(e*x + d)*sqrt(-d)), 2/15*(15*(c^3*d^3*e^2*x^2 + 2*c^3*d^4*e*x + c^3*d^
5)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d
- b*e))/(sqrt(e*x + d)*c)) + 15*(c^3*d^5 - 3*b*c^2*d^4*e + 3*b^2*c*d^3*e^2 - b^3
*d^2*e^3 + (c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*c*d*e^4 - b^3*e^5)*x^2 + 2*(c^
3*d^4*e - 3*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*sqrt(e*x + d)*arctan
(d/(sqrt(e*x + d)*sqrt(-d))) - (58*b*c^2*d^4*e - 66*b^2*c*d^3*e^2 + 23*b^3*d^2*e
^3 + 15*(3*b*c^2*d^2*e^3 - 3*b^2*c*d*e^4 + b^3*e^5)*x^2 + 5*(20*b*c^2*d^3*e^2 -
21*b^2*c*d^2*e^3 + 7*b^3*d*e^4)*x)*sqrt(-d))/((b*c^3*d^8 - 3*b^2*c^2*d^7*e + 3*b
^3*c*d^6*e^2 - b^4*d^5*e^3 + (b*c^3*d^6*e^2 - 3*b^2*c^2*d^5*e^3 + 3*b^3*c*d^4*e^
4 - b^4*d^3*e^5)*x^2 + 2*(b*c^3*d^7*e - 3*b^2*c^2*d^6*e^2 + 3*b^3*c*d^5*e^3 - b^
4*d^4*e^4)*x)*sqrt(e*x + d)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (b + c x\right ) \left (d + e x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(b + c*x)*(d + e*x)**(7/2)), x)

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GIAC/XCAS [A]  time = 0.214661, size = 389, normalized size = 2.08 \[ -\frac{2 \, c^{4} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (45 \,{\left (x e + d\right )}^{2} c^{2} d^{2} e + 10 \,{\left (x e + d\right )} c^{2} d^{3} e + 3 \, c^{2} d^{4} e - 45 \,{\left (x e + d\right )}^{2} b c d e^{2} - 15 \,{\left (x e + d\right )} b c d^{2} e^{2} - 6 \, b c d^{3} e^{2} + 15 \,{\left (x e + d\right )}^{2} b^{2} e^{3} + 5 \,{\left (x e + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3}\right )}}{15 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} + \frac{2 \, \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c^4*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^3*d^3 - 3*b^2*c^2*d^2*
e + 3*b^3*c*d*e^2 - b^4*e^3)*sqrt(-c^2*d + b*c*e)) - 2/15*(45*(x*e + d)^2*c^2*d^
2*e + 10*(x*e + d)*c^2*d^3*e + 3*c^2*d^4*e - 45*(x*e + d)^2*b*c*d*e^2 - 15*(x*e
+ d)*b*c*d^2*e^2 - 6*b*c*d^3*e^2 + 15*(x*e + d)^2*b^2*e^3 + 5*(x*e + d)*b^2*d*e^
3 + 3*b^2*d^2*e^3)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*(x
*e + d)^(5/2)) + 2*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)*d^3)