3.426 \(\int \frac{x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=320 \[ \frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \]

[Out]

-(x^(5/2)*(A + B*x))/(4*c*(a + c*x^2)^2) - (Sqrt[x]*(5*A + 7*B*x))/(16*c^2*(a +
c*x^2)) - ((21*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(
1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[1 +
(Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]
*B - 5*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(6
4*Sqrt[2]*a^(3/4)*c^(11/4)) - ((21*Sqrt[a]*B - 5*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(3/4)*c^(11/4))

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Rubi [A]  time = 0.607855, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x))/(a + c*x^2)^3,x]

[Out]

-(x^(5/2)*(A + B*x))/(4*c*(a + c*x^2)^2) - (Sqrt[x]*(5*A + 7*B*x))/(16*c^2*(a +
c*x^2)) - ((21*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(
1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[1 +
(Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]
*B - 5*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(6
4*Sqrt[2]*a^(3/4)*c^(11/4)) - ((21*Sqrt[a]*B - 5*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(3/4)*c^(11/4))

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Rubi in Sympy [A]  time = 108.207, size = 304, normalized size = 0.95 \[ - \frac{x^{\frac{5}{2}} \left (2 A + 2 B x\right )}{8 c \left (a + c x^{2}\right )^{2}} - \frac{\sqrt{x} \left (10 A + 14 B x\right )}{32 c^{2} \left (a + c x^{2}\right )} - \frac{\sqrt{2} \left (5 A \sqrt{c} - 21 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{3}{4}} c^{\frac{11}{4}}} + \frac{\sqrt{2} \left (5 A \sqrt{c} - 21 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{3}{4}} c^{\frac{11}{4}}} - \frac{\sqrt{2} \left (5 A \sqrt{c} + 21 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{3}{4}} c^{\frac{11}{4}}} + \frac{\sqrt{2} \left (5 A \sqrt{c} + 21 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{3}{4}} c^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)/(c*x**2+a)**3,x)

[Out]

-x**(5/2)*(2*A + 2*B*x)/(8*c*(a + c*x**2)**2) - sqrt(x)*(10*A + 14*B*x)/(32*c**2
*(a + c*x**2)) - sqrt(2)*(5*A*sqrt(c) - 21*B*sqrt(a))*log(-sqrt(2)*a**(1/4)*c**(
3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(128*a**(3/4)*c**(11/4)) + sqrt(2)*(5*A*sq
rt(c) - 21*B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) +
c*x)/(128*a**(3/4)*c**(11/4)) - sqrt(2)*(5*A*sqrt(c) + 21*B*sqrt(a))*atan(1 - sq
rt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(3/4)*c**(11/4)) + sqrt(2)*(5*A*sqrt(c)
+ 21*B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(3/4)*c**(11/
4))

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Mathematica [A]  time = 0.540945, size = 301, normalized size = 0.94 \[ \frac{\frac{\sqrt{2} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{3/4}}-\frac{\sqrt{2} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{3/4}}-\frac{2 \sqrt{2} \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{2 \sqrt{2} \left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}+\frac{32 a c^{3/4} \sqrt{x} (A+B x)}{\left (a+c x^2\right )^2}-\frac{8 c^{3/4} \sqrt{x} (9 A+11 B x)}{a+c x^2}}{128 c^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x))/(a + c*x^2)^3,x]

[Out]

((32*a*c^(3/4)*Sqrt[x]*(A + B*x))/(a + c*x^2)^2 - (8*c^(3/4)*Sqrt[x]*(9*A + 11*B
*x))/(a + c*x^2) - (2*Sqrt[2]*(21*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c
^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (2*Sqrt[2]*(21*Sqrt[a]*B + 5*A*Sqrt[c])*ArcT
an[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (Sqrt[2]*(21*Sqrt[a]*B - 5*
A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/a^(3/4) -
(Sqrt[2]*(21*Sqrt[a]*B - 5*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqr
t[x] + Sqrt[c]*x])/a^(3/4))/(128*c^(11/4))

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Maple [A]  time = 0.027, size = 327, normalized size = 1. \[ 2\,{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ( -{\frac{11\,B{x}^{7/2}}{32\,c}}-{\frac{9\,A{x}^{5/2}}{32\,c}}-{\frac{7\,aB{x}^{3/2}}{32\,{c}^{2}}}-{\frac{5\,aA\sqrt{x}}{32\,{c}^{2}}} \right ) }+{\frac{5\,A\sqrt{2}}{128\,a{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{5\,A\sqrt{2}}{64\,a{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{5\,A\sqrt{2}}{64\,a{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{21\,B\sqrt{2}}{128\,{c}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,B\sqrt{2}}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,B\sqrt{2}}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)/(c*x^2+a)^3,x)

[Out]

2*(-11/32*B*x^(7/2)/c-9/32/c*A*x^(5/2)-7/32*B*a/c^2*x^(3/2)-5/32*a*A/c^2*x^(1/2)
)/(c*x^2+a)^2+5/128*A/c^2*(a/c)^(1/4)/a*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)))+5/64*A/c^2*(a/c)^(1/
4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+5/64*A/c^2*(a/c)^(1/4)/a*2^(1
/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+21/128*B/c^3/(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)))+21/64*B/c^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+21/
64*B/c^3/(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^(7/2)/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.341789, size = 1331, normalized size = 4.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(c*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/64*((c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)*sqrt(-(a*c^5*sqrt(-(194481*B^4*a^2 - 220
50*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 210*A*B)/(a*c^5))*log(-(194481*B^4*a
^2 - 625*A^4*c^2)*sqrt(x) + (21*B*a^3*c^8*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*
a*c + 625*A^4*c^2)/(a^3*c^11)) - 2205*A*B^2*a^2*c^3 + 125*A^3*a*c^4)*sqrt(-(a*c^
5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 210*A*B
)/(a*c^5))) - (c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)*sqrt(-(a*c^5*sqrt(-(194481*B^4*a
^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 210*A*B)/(a*c^5))*log(-(1944
81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (21*B*a^3*c^8*sqrt(-(194481*B^4*a^2 - 22050*
A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) - 2205*A*B^2*a^2*c^3 + 125*A^3*a*c^4)*sqr
t(-(a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) +
210*A*B)/(a*c^5))) - (c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)*sqrt((a*c^5*sqrt(-(19448
1*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) - 210*A*B)/(a*c^5))*log
(-(194481*B^4*a^2 - 625*A^4*c^2)*sqrt(x) + (21*B*a^3*c^8*sqrt(-(194481*B^4*a^2 -
22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 2205*A*B^2*a^2*c^3 - 125*A^3*a*c
^4)*sqrt((a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^
11)) - 210*A*B)/(a*c^5))) + (c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)*sqrt((a*c^5*sqrt(-
(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) - 210*A*B)/(a*c^5
))*log(-(194481*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (21*B*a^3*c^8*sqrt(-(194481*B^4
*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 2205*A*B^2*a^2*c^3 - 125*A
^3*a*c^4)*sqrt((a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(
a^3*c^11)) - 210*A*B)/(a*c^5))) - 4*(11*B*c*x^3 + 9*A*c*x^2 + 7*B*a*x + 5*A*a)*s
qrt(x))/(c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)

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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**(7/2)*(B*x+A)/(c*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284006, size = 396, normalized size = 1.24 \[ -\frac{11 \, B c x^{\frac{7}{2}} + 9 \, A c x^{\frac{5}{2}} + 7 \, B a x^{\frac{3}{2}} + 5 \, A a \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} c^{2}} + \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a c^{5}} + \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a c^{5}} + \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a c^{5}} - \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(7/2)/(c*x^2 + a)^3,x, algorithm="giac")

[Out]

-1/16*(11*B*c*x^(7/2) + 9*A*c*x^(5/2) + 7*B*a*x^(3/2) + 5*A*a*sqrt(x))/((c*x^2 +
a)^2*c^2) + 1/64*sqrt(2)*(5*(a*c^3)^(1/4)*A*c^2 + 21*(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*c^5) + 1/64*sqrt(2)*
(5*(a*c^3)^(1/4)*A*c^2 + 21*(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*c^5) + 1/128*sqrt(2)*(5*(a*c^3)^(1/4)*A*c^2 -
21*(a*c^3)^(3/4)*B)*ln(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a*c^5) - 1
/128*sqrt(2)*(5*(a*c^3)^(1/4)*A*c^2 - 21*(a*c^3)^(3/4)*B)*ln(-sqrt(2)*sqrt(x)*(a
/c)^(1/4) + x + sqrt(a/c))/(a*c^5)