3.428 \(\int \frac{x^{3/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=315 \[ \frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac{\sqrt{x} (A+3 B x)}{16 a c \left (a+c x^2\right )} \]

[Out]

-(Sqrt[x]*(A + B*x))/(4*c*(a + c*x^2)^2) + (Sqrt[x]*(A + 3*B*x))/(16*a*c*(a + c*
x^2)) - (3*(Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]
)/(32*Sqrt[2]*a^(7/4)*c^(7/4)) + (3*(Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*
c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(7/4)*c^(7/4)) + (3*(Sqrt[a]*B - A*Sqrt
[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(
7/4)*c^(7/4)) - (3*(Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(7/4)*c^(7/4))

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Rubi [A]  time = 0.589067, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ \frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac{\sqrt{x} (A+3 B x)}{16 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^(3/2)*(A + B*x))/(a + c*x^2)^3,x]

[Out]

-(Sqrt[x]*(A + B*x))/(4*c*(a + c*x^2)^2) + (Sqrt[x]*(A + 3*B*x))/(16*a*c*(a + c*
x^2)) - (3*(Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]
)/(32*Sqrt[2]*a^(7/4)*c^(7/4)) + (3*(Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*
c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(7/4)*c^(7/4)) + (3*(Sqrt[a]*B - A*Sqrt
[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(
7/4)*c^(7/4)) - (3*(Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(7/4)*c^(7/4))

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Rubi in Sympy [A]  time = 112.037, size = 296, normalized size = 0.94 \[ - \frac{\sqrt{x} \left (2 A + 2 B x\right )}{8 c \left (a + c x^{2}\right )^{2}} + \frac{\sqrt{x} \left (A + 3 B x\right )}{16 a c \left (a + c x^{2}\right )} - \frac{3 \sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{7}{4}} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{7}{4}} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)/(c*x**2+a)**3,x)

[Out]

-sqrt(x)*(2*A + 2*B*x)/(8*c*(a + c*x**2)**2) + sqrt(x)*(A + 3*B*x)/(16*a*c*(a +
c*x**2)) - 3*sqrt(2)*(A*sqrt(c) - B*sqrt(a))*log(-sqrt(2)*a**(1/4)*c**(3/4)*sqrt
(x) + sqrt(a)*sqrt(c) + c*x)/(128*a**(7/4)*c**(7/4)) + 3*sqrt(2)*(A*sqrt(c) - B*
sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(128*a**
(7/4)*c**(7/4)) - 3*sqrt(2)*(A*sqrt(c) + B*sqrt(a))*atan(1 - sqrt(2)*c**(1/4)*sq
rt(x)/a**(1/4))/(64*a**(7/4)*c**(7/4)) + 3*sqrt(2)*(A*sqrt(c) + B*sqrt(a))*atan(
1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(7/4)*c**(7/4))

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Mathematica [A]  time = 0.580643, size = 296, normalized size = 0.94 \[ \frac{\frac{3 \sqrt{2} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{7/4}}-\frac{3 \sqrt{2} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{7/4}}-\frac{6 \sqrt{2} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{2} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{32 c^{3/4} \sqrt{x} (A+B x)}{\left (a+c x^2\right )^2}+\frac{8 c^{3/4} \sqrt{x} (A+3 B x)}{a \left (a+c x^2\right )}}{128 c^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(3/2)*(A + B*x))/(a + c*x^2)^3,x]

[Out]

((-32*c^(3/4)*Sqrt[x]*(A + B*x))/(a + c*x^2)^2 + (8*c^(3/4)*Sqrt[x]*(A + 3*B*x))
/(a*(a + c*x^2)) - (6*Sqrt[2]*(Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4
)*Sqrt[x])/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*(Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 + (Sq
rt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/a^(7/4) + (3*Sqrt[2]*(Sqrt[a]*B - A*Sqrt[c])*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/a^(7/4) - (3*Sqrt[2]*(
Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/a^(7/4))/(128*c^(7/4))

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Maple [A]  time = 0.025, size = 334, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{3\,B{x}^{7/2}}{32\,a}}+1/32\,{\frac{A{x}^{5/2}}{a}}-1/32\,{\frac{B{x}^{3/2}}{c}}-{\frac{3\,A\sqrt{x}}{32\,c}} \right ) }+{\frac{3\,A\sqrt{2}}{128\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,A\sqrt{2}}{64\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,A\sqrt{2}}{64\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{3\,B\sqrt{2}}{128\,a{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,B\sqrt{2}}{64\,a{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,B\sqrt{2}}{64\,a{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)/(c*x^2+a)^3,x)

[Out]

2*(3/32*B/a*x^(7/2)+1/32*A/a*x^(5/2)-1/32*B*x^(3/2)/c-3/32*A*x^(1/2)/c)/(c*x^2+a
)^2+3/128*A/c/a^2*(a/c)^(1/4)*2^(1/2)*ln((x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1
/2))/(x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))+3/64*A/c/a^2*(a/c)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+3/64*A/c/a^2*(a/c)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+3/128*B/c^2/a/(a/c)^(1/4)*2^(1/2)*ln((x-(a/c)^(1
/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))+3/
64*B/c^2/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+3/64*B/c^2/
a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)

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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((B*x + A)*x^(3/2)/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.318202, size = 1326, normalized size = 4.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(3/2)/(c*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/64*(3*(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)*sqrt(-(a^3*c^3*sqrt(-(B^4*a^2 - 2*A^
2*B^2*a*c + A^4*c^2)/(a^7*c^7)) + 2*A*B)/(a^3*c^3))*log(-27*(B^4*a^2 - A^4*c^2)*
sqrt(x) + 27*(B*a^6*c^5*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) - A
*B^2*a^3*c^2 + A^3*a^2*c^3)*sqrt(-(a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*
c^2)/(a^7*c^7)) + 2*A*B)/(a^3*c^3))) - 3*(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)*sqr
t(-(a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) + 2*A*B)/(a^3*c
^3))*log(-27*(B^4*a^2 - A^4*c^2)*sqrt(x) - 27*(B*a^6*c^5*sqrt(-(B^4*a^2 - 2*A^2*
B^2*a*c + A^4*c^2)/(a^7*c^7)) - A*B^2*a^3*c^2 + A^3*a^2*c^3)*sqrt(-(a^3*c^3*sqrt
(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) + 2*A*B)/(a^3*c^3))) - 3*(a*c^3
*x^4 + 2*a^2*c^2*x^2 + a^3*c)*sqrt((a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4
*c^2)/(a^7*c^7)) - 2*A*B)/(a^3*c^3))*log(-27*(B^4*a^2 - A^4*c^2)*sqrt(x) + 27*(B
*a^6*c^5*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) + A*B^2*a^3*c^2 -
A^3*a^2*c^3)*sqrt((a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7))
- 2*A*B)/(a^3*c^3))) + 3*(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)*sqrt((a^3*c^3*sqrt(
-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^7*c^7)) - 2*A*B)/(a^3*c^3))*log(-27*(B^4
*a^2 - A^4*c^2)*sqrt(x) - 27*(B*a^6*c^5*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2
)/(a^7*c^7)) + A*B^2*a^3*c^2 - A^3*a^2*c^3)*sqrt((a^3*c^3*sqrt(-(B^4*a^2 - 2*A^2
*B^2*a*c + A^4*c^2)/(a^7*c^7)) - 2*A*B)/(a^3*c^3))) + 4*(3*B*c*x^3 + A*c*x^2 - B
*a*x - 3*A*a)*sqrt(x))/(a*c^3*x^4 + 2*a^2*c^2*x^2 + a^3*c)

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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(x**(3/2)*(B*x+A)/(c*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284131, size = 390, normalized size = 1.24 \[ \frac{3 \, B c x^{\frac{7}{2}} + A c x^{\frac{5}{2}} - B a x^{\frac{3}{2}} - 3 \, A a \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} a c} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{2} c^{4}} - \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{2} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(3/2)/(c*x^2 + a)^3,x, algorithm="giac")

[Out]

1/16*(3*B*c*x^(7/2) + A*c*x^(5/2) - B*a*x^(3/2) - 3*A*a*sqrt(x))/((c*x^2 + a)^2*
a*c) + 3/64*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (a*c^3)^(3/4)*B)*arctan(1/2*sqrt(2)*(
sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^2*c^4) + 3/64*sqrt(2)*((a*c^3)^
(1/4)*A*c^2 + (a*c^3)^(3/4)*B)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt
(x))/(a/c)^(1/4))/(a^2*c^4) + 3/128*sqrt(2)*((a*c^3)^(1/4)*A*c^2 - (a*c^3)^(3/4)
*B)*ln(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^4) - 3/128*sqrt(2)*((
a*c^3)^(1/4)*A*c^2 - (a*c^3)^(3/4)*B)*ln(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt
(a/c))/(a^2*c^4)