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

Optimal. Leaf size=191 \[ -\frac{2 (d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{e \sqrt{a+c x^2} \left (4 \left (-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )+c d e x \left (7 a e^2+2 c d^2\right )\right )}{3 a^2 c^3}+\frac{5 d e^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{5/2}}-\frac{(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.501563, antiderivative size = 191, 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 \[ -\frac{2 (d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{e \sqrt{a+c x^2} \left (4 \left (-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )+c d e x \left (7 a e^2+2 c d^2\right )\right )}{3 a^2 c^3}+\frac{5 d e^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{5/2}}-\frac{(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 59.1796, size = 184, normalized size = 0.96 \[ \frac{5 d e^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{5}{2}}} - \frac{\left (d + e x\right )^{4} \left (a e - c d x\right )}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} - \frac{\left (d + e x\right )^{2} \left (8 a^{2} e^{3} - 4 c d x \left (3 a e^{2} + c d^{2}\right )\right )}{6 a^{2} c^{2} \sqrt{a + c x^{2}}} + \frac{e \sqrt{a + c x^{2}} \left (32 a^{2} e^{4} - 64 a c d^{2} e^{2} - 16 c^{2} d^{4} - 4 c d e x \left (7 a e^{2} + 2 c d^{2}\right )\right )}{12 a^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*d*e**4*atanh(sqrt(c)*x/sqrt(a + c*x**2))/c**(5/2) - (d + e*x)**4*(a*e - c*d*x)
/(3*a*c*(a + c*x**2)**(3/2)) - (d + e*x)**2*(8*a**2*e**3 - 4*c*d*x*(3*a*e**2 + c
*d**2))/(6*a**2*c**2*sqrt(a + c*x**2)) + e*sqrt(a + c*x**2)*(32*a**2*e**4 - 64*a
*c*d**2*e**2 - 16*c**2*d**4 - 4*c*d*e*x*(7*a*e**2 + 2*c*d**2))/(12*a**2*c**3)

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Mathematica [A]  time = 0.375995, size = 167, normalized size = 0.87 \[ \frac{8 a^4 e^5+a^3 c e^3 \left (-20 d^2-15 d e x+12 e^2 x^2\right )+a^2 c^2 e \left (-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right )+a c^3 d^3 x \left (3 d^2+10 e^2 x^2\right )+2 c^4 d^5 x^3}{3 a^2 c^3 \left (a+c x^2\right )^{3/2}}+\frac{5 d e^4 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*(3*d^2 + 10*e^2*x^2) + a^3*c*e^3*(-20*d
^2 - 15*d*e*x + 12*e^2*x^2) + a^2*c^2*e*(-5*d^4 - 30*d^2*e^2*x^2 - 20*d*e^3*x^3
+ 3*e^4*x^4))/(3*a^2*c^3*(a + c*x^2)^(3/2)) + (5*d*e^4*Log[c*x + Sqrt[c]*Sqrt[a
+ c*x^2]])/c^(5/2)

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Maple [A]  time = 0.021, size = 270, normalized size = 1.4 \[{\frac{{d}^{5}x}{3\,a} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,{d}^{5}x}{3\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{e}^{5}{x}^{4}}{c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{a{e}^{5}{x}^{2}}{{c}^{2} \left ( c{x}^{2}+a \right ) ^{3/2}}}+{\frac{8\,{e}^{5}{a}^{2}}{3\,{c}^{3}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,d{e}^{4}{x}^{3}}{3\,c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-5\,{\frac{d{e}^{4}x}{{c}^{2}\sqrt{c{x}^{2}+a}}}+5\,{\frac{d{e}^{4}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }{{c}^{5/2}}}-10\,{\frac{{d}^{2}{e}^{3}{x}^{2}}{c \left ( c{x}^{2}+a \right ) ^{3/2}}}-{\frac{20\,{d}^{2}{e}^{3}a}{3\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{10\,{d}^{3}{e}^{2}x}{3\,c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{10\,{d}^{3}{e}^{2}x}{3\,ac}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{5\,{d}^{4}e}{3\,c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*d^5*x/a/(c*x^2+a)^(3/2)+2/3*d^5/a^2*x/(c*x^2+a)^(1/2)+e^5*x^4/c/(c*x^2+a)^(3
/2)+4*e^5*a/c^2*x^2/(c*x^2+a)^(3/2)+8/3*e^5*a^2/c^3/(c*x^2+a)^(3/2)-5/3*d*e^4*x^
3/c/(c*x^2+a)^(3/2)-5*d*e^4/c^2*x/(c*x^2+a)^(1/2)+5*d*e^4/c^(5/2)*ln(c^(1/2)*x+(
c*x^2+a)^(1/2))-10*d^2*e^3*x^2/c/(c*x^2+a)^(3/2)-20/3*d^2*e^3*a/c^2/(c*x^2+a)^(3
/2)-10/3*d^3*e^2/c*x/(c*x^2+a)^(3/2)+10/3*d^3*e^2/a/c*x/(c*x^2+a)^(1/2)-5/3*d^4*
e/c/(c*x^2+a)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251985, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \,{\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \,{\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \,{\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{c} + 15 \,{\left (a^{2} c^{3} d e^{4} x^{4} + 2 \, a^{3} c^{2} d e^{4} x^{2} + a^{4} c d e^{4}\right )} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right )}{6 \,{\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )} \sqrt{c}}, \frac{{\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \,{\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \,{\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \,{\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt{c x^{2} + a} \sqrt{-c} + 15 \,{\left (a^{2} c^{3} d e^{4} x^{4} + 2 \, a^{3} c^{2} d e^{4} x^{2} + a^{4} c d e^{4}\right )} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right )}{3 \,{\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(2*(3*a^2*c^2*e^5*x^4 - 5*a^2*c^2*d^4*e - 20*a^3*c*d^2*e^3 + 8*a^4*e^5 + 2*
(c^4*d^5 + 5*a*c^3*d^3*e^2 - 10*a^2*c^2*d*e^4)*x^3 - 6*(5*a^2*c^2*d^2*e^3 - 2*a^
3*c*e^5)*x^2 + 3*(a*c^3*d^5 - 5*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a)*sqrt(c) + 15*(a^
2*c^3*d*e^4*x^4 + 2*a^3*c^2*d*e^4*x^2 + a^4*c*d*e^4)*log(-2*sqrt(c*x^2 + a)*c*x
- (2*c*x^2 + a)*sqrt(c)))/((a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3)*sqrt(c)), 1/3
*((3*a^2*c^2*e^5*x^4 - 5*a^2*c^2*d^4*e - 20*a^3*c*d^2*e^3 + 8*a^4*e^5 + 2*(c^4*d
^5 + 5*a*c^3*d^3*e^2 - 10*a^2*c^2*d*e^4)*x^3 - 6*(5*a^2*c^2*d^2*e^3 - 2*a^3*c*e^
5)*x^2 + 3*(a*c^3*d^5 - 5*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a)*sqrt(-c) + 15*(a^2*c^3
*d*e^4*x^4 + 2*a^3*c^2*d*e^4*x^2 + a^4*c*d*e^4)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a
)))/((a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222821, size = 269, normalized size = 1.41 \[ -\frac{5 \, d e^{4}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{5}{2}}} + \frac{{\left ({\left (x{\left (\frac{3 \, x e^{5}}{c} + \frac{2 \,{\left (c^{6} d^{5} + 5 \, a c^{5} d^{3} e^{2} - 10 \, a^{2} c^{4} d e^{4}\right )}}{a^{2} c^{5}}\right )} - \frac{6 \,{\left (5 \, a^{2} c^{4} d^{2} e^{3} - 2 \, a^{3} c^{3} e^{5}\right )}}{a^{2} c^{5}}\right )} x + \frac{3 \,{\left (a c^{5} d^{5} - 5 \, a^{3} c^{3} d e^{4}\right )}}{a^{2} c^{5}}\right )} x - \frac{5 \, a^{2} c^{4} d^{4} e + 20 \, a^{3} c^{3} d^{2} e^{3} - 8 \, a^{4} c^{2} e^{5}}{a^{2} c^{5}}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*d*e^4*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/3*(((x*(3*x*e^5/c + 2
*(c^6*d^5 + 5*a*c^5*d^3*e^2 - 10*a^2*c^4*d*e^4)/(a^2*c^5)) - 6*(5*a^2*c^4*d^2*e^
3 - 2*a^3*c^3*e^5)/(a^2*c^5))*x + 3*(a*c^5*d^5 - 5*a^3*c^3*d*e^4)/(a^2*c^5))*x -
(5*a^2*c^4*d^4*e + 20*a^3*c^3*d^2*e^3 - 8*a^4*c^2*e^5)/(a^2*c^5))/(c*x^2 + a)^(
3/2)