3.571 \(\int \frac{1}{(d+e x) \left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=154 \[ \frac{3 a^2 e^3+c d x \left (5 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac{e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.346136, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{3 a^2 e^3+c d x \left (5 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac{e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + c*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 50.8832, size = 136, normalized size = 0.88 \[ - \frac{e^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{5}{2}}} + \frac{a e + c d x}{3 a \left (a + c x^{2}\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} + \frac{3 a^{2} e^{3} + c d x \left (5 a e^{2} + 2 c d^{2}\right )}{3 a^{2} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.4722, size = 171, normalized size = 1.11 \[ \frac{\left (a+c x^2\right ) \left (3 a^2 e^3+5 a c d e^2 x+2 c^2 d^3 x\right )+a \left (a e^2+c d^2\right ) (a e+c d x)}{3 a^2 \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )^2}-\frac{e^4 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{5/2}}+\frac{e^4 \log (d+e x)}{\left (a e^2+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + c*x^2)^(5/2)),x]

[Out]

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

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Maple [B]  time = 0.014, size = 454, normalized size = 3. \[{\frac{e}{3\,a{e}^{2}+3\,c{d}^{2}} \left ( c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}} \right ) ^{-{\frac{3}{2}}}}+{\frac{cdx}{ \left ( 3\,a{e}^{2}+3\,c{d}^{2} \right ) a} \left ( c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,cdx}{ \left ( 3\,a{e}^{2}+3\,c{d}^{2} \right ){a}^{2}}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{e}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{d{e}^{2}cx}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{e}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+a)^(5/2),x)

[Out]

1/3*e/(a*e^2+c*d^2)/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+1/3*c*
d/(a*e^2+c*d^2)/a/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x+2/3*c*
d/(a*e^2+c*d^2)/a^2/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+e^3/
(a*e^2+c*d^2)^2/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+e^2/(a*e^2
+c*d^2)^2*d/a/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*c*x-e^3/(a*e
^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*
((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))
/(d/e+x))

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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 + a)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.357909, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, a^{2} c e^{3} x^{2} + a^{2} c d^{2} e + 4 \, a^{3} e^{3} +{\left (2 \, c^{3} d^{3} + 5 \, a c^{2} d e^{2}\right )} x^{3} + 3 \,{\left (a c^{2} d^{3} + 2 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} + 3 \,{\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{6 \,{\left (a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4} +{\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{4} + 2 \,{\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}}}, \frac{{\left (3 \, a^{2} c e^{3} x^{2} + a^{2} c d^{2} e + 4 \, a^{3} e^{3} +{\left (2 \, c^{3} d^{3} + 5 \, a c^{2} d e^{2}\right )} x^{3} + 3 \,{\left (a c^{2} d^{3} + 2 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} + 3 \,{\left (a^{2} c^{2} e^{4} x^{4} + 2 \, a^{3} c e^{4} x^{2} + a^{4} e^{4}\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{3 \,{\left (a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4} +{\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{4} + 2 \,{\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4}\right )} x^{2}\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

[1/6*(2*(3*a^2*c*e^3*x^2 + a^2*c*d^2*e + 4*a^3*e^3 + (2*c^3*d^3 + 5*a*c^2*d*e^2)
*x^3 + 3*(a*c^2*d^3 + 2*a^2*c*d*e^2)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 3*
(a^2*c^2*e^4*x^4 + 2*a^3*c*e^4*x^2 + a^4*e^4)*log(((2*a*c*d*e*x - a*c*d^2 - 2*a^
2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2)*sqrt(c*d^2 + a*e^2) + 2*(a*c*d^2*e + a^2*e^3
- (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((a^4*c^
2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e
^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(c*d^2 + a*e^
2)), 1/3*((3*a^2*c*e^3*x^2 + a^2*c*d^2*e + 4*a^3*e^3 + (2*c^3*d^3 + 5*a*c^2*d*e^
2)*x^3 + 3*(a*c^2*d^3 + 2*a^2*c*d*e^2)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) +
3*(a^2*c^2*e^4*x^4 + 2*a^3*c*e^4*x^2 + a^4*e^4)*arctan(sqrt(-c*d^2 - a*e^2)*(c*
d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 +
a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4
+ 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(-c*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.225761, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

Done