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

Optimal. Leaf size=351 \[ \frac{11 (17 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}+\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2} \]

[Out]

(-11*(17*A*b - 5*a*B))/(180*a^3*b*x^(5/2)) + (A*b - a*B)/(6*a*b*x^(5/2)*(a + b*x
^3)^2) + (17*A*b - 5*a*B)/(36*a^2*b*x^(5/2)*(a + b*x^3)) + (11*(17*A*b - 5*a*B)*
ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)*b^(1/6)) - (11*(17*
A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)*b^(1/6
)) - (11*(17*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(23/6)*b^(1/
6)) + (11*(17*A*b - 5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/
3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6)) - (11*(17*A*b - 5*a*B)*Log[a^(1/3) + Sqrt[
3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6))

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Rubi [A]  time = 1.20533, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{11 (17 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{23/6} \sqrt [6]{b}}+\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{23/6} \sqrt [6]{b}}-\frac{11 (17 A b-5 a B)}{180 a^3 b x^{5/2}}+\frac{17 A b-5 a B}{36 a^2 b x^{5/2} \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^{5/2} \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-11*(17*A*b - 5*a*B))/(180*a^3*b*x^(5/2)) + (A*b - a*B)/(6*a*b*x^(5/2)*(a + b*x
^3)^2) + (17*A*b - 5*a*B)/(36*a^2*b*x^(5/2)*(a + b*x^3)) + (11*(17*A*b - 5*a*B)*
ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)*b^(1/6)) - (11*(17*
A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(23/6)*b^(1/6
)) - (11*(17*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(23/6)*b^(1/
6)) + (11*(17*A*b - 5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/
3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6)) - (11*(17*A*b - 5*a*B)*Log[a^(1/3) + Sqrt[
3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(23/6)*b^(1/6))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.735093, size = 308, normalized size = 0.88 \[ \frac{\frac{360 a^{11/6} \sqrt{x} (a B-A b)}{\left (a+b x^3\right )^2}+\frac{60 a^{5/6} \sqrt{x} (11 a B-23 A b)}{a+b x^3}-\frac{864 a^{5/6} A}{x^{5/2}}+\frac{55 \sqrt{3} (17 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [6]{b}}+\frac{55 \sqrt{3} (5 a B-17 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [6]{b}}+\frac{110 (17 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac{110 (17 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt [6]{b}}+\frac{220 (5 a B-17 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}}{2160 a^{23/6}} \]

Antiderivative was successfully verified.

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

[Out]

((-864*a^(5/6)*A)/x^(5/2) + (360*a^(11/6)*(-(A*b) + a*B)*Sqrt[x])/(a + b*x^3)^2
+ (60*a^(5/6)*(-23*A*b + 11*a*B)*Sqrt[x])/(a + b*x^3) + (110*(17*A*b - 5*a*B)*Ar
cTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(1/6) - (110*(17*A*b - 5*a*B)*Arc
Tan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(1/6) + (220*(-17*A*b + 5*a*B)*Arc
Tan[(b^(1/6)*Sqrt[x])/a^(1/6)])/b^(1/6) + (55*Sqrt[3]*(17*A*b - 5*a*B)*Log[a^(1/
3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/b^(1/6) + (55*Sqrt[3]*(-17*A*
b + 5*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/b^(1/6))/
(2160*a^(23/6))

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Maple [A]  time = 0.068, size = 429, normalized size = 1.2 \[ -{\frac{2\,A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{23\,{b}^{2}A}{36\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{11\,Bb}{36\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{29\,Ab}{36\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}\sqrt{x}}+{\frac{17\,B}{36\,a \left ( b{x}^{3}+a \right ) ^{2}}\sqrt{x}}-{\frac{187\,Ab}{108\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{187\,Ab\sqrt{3}}{432\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{187\,Ab}{216\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{187\,Ab\sqrt{3}}{432\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{187\,Ab}{216\,{a}^{4}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{55\,B}{108\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{55\,B\sqrt{3}}{432\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{55\,B}{216\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{55\,B\sqrt{3}}{432\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{55\,B}{216\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*A/a^3/x^(5/2)-23/36/a^3/(b*x^3+a)^2*x^(7/2)*b^2*A+11/36/a^2/(b*x^3+a)^2*x^(
7/2)*b*B-29/36/a^2/(b*x^3+a)^2*A*x^(1/2)*b+17/36/a/(b*x^3+a)^2*B*x^(1/2)-187/108
/a^4*A*b*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))+187/432/a^4*A*b*3^(1/2)*(a/b)^(
1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-187/216/a^4*A*b*(a/b)^(1/6)*a
rctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))-187/432/a^4*A*b*3^(1/2)*(a/b)^(1/6)*ln(x+3
^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-187/216/a^4*A*b*(a/b)^(1/6)*arctan(2*x^(
1/2)/(a/b)^(1/6)+3^(1/2))+55/108/a^3*B*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))-5
5/432/a^3*B*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+55
/216/a^3*B*(a/b)^(1/6)*arctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))+55/432/a^3*B*3^(1/
2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+55/216/a^3*B*(a/b)^
(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))

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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((B*x^3 + A)/((b*x^3 + a)^3*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290089, size = 3190, normalized size = 9.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2160*(132*(5*B*a*b - 17*A*b^2)*x^6 + 204*(5*B*a^2 - 17*A*a*b)*x^3 + 220*sqrt(3
)*(a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*
a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^
2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*arctan(-sqrt(3)
*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*
A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6
*b^6)/(a^23*b))^(1/6)/(a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B
^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*
B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6) + 2*(5*B*a - 17*A*b)*sqrt(x) - 2*sqr
t(a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500
*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^
6*b^6)/(a^23*b))^(1/3) + (25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x + (5*B*a^5 -
17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4
*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^
5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)))) + 220*sqrt(3)*(a^3*b^2*x^8 + 2*a^4*b*x^
5 + a^5*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4
*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^
5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*arctan(sqrt(3)*a^4*(-(15625*B^6*a^6 - 3187
50*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A
^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)/(a^4*(
-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^
3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/
(a^23*b))^(1/6) - 2*(5*B*a - 17*A*b)*sqrt(x) + 2*sqrt(a^8*(-(15625*B^6*a^6 - 318
750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*
A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/3) + (25
*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x - (5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(1562
5*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*
b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*
b))^(1/6)))) - 864*A*a^2 + 55*(a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x^2)*sqrt(x)*(-(1
5625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a
^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^
23*b))^(1/6)*log(121*a^8*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4
*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*
a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/3) + 121*(25*B^2*a^2 - 170*A*B*a*b + 289*
A^2*b^2)*x + 121*(5*B*a^5 - 17*A*a^4*b)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*
a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^
2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)) - 55*(a^3*b^2*
x^8 + 2*a^4*b*x^5 + a^5*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 270
9375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 425
95710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*log(121*a^8*(-(15625*B^6*a
^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 3
1320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/
3) + 121*(25*B^2*a^2 - 170*A*B*a*b + 289*A^2*b^2)*x - 121*(5*B*a^5 - 17*A*a^4*b)
*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282
500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569
*A^6*b^6)/(a^23*b))^(1/6)) - 110*(a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x^2)*sqrt(x)*(
-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^
3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/
(a^23*b))^(1/6)*log(11*a^4*(-(15625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B
^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*
B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6) - 11*(5*B*a - 17*A*b)*sqrt(x)) + 110
*(a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 318750*A*B^5*a
^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a^3*b^3 + 31320375*A^4*B^2*a^2
*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^23*b))^(1/6)*log(-11*a^4*(-(1
5625*B^6*a^6 - 318750*A*B^5*a^5*b + 2709375*A^2*B^4*a^4*b^2 - 12282500*A^3*B^3*a
^3*b^3 + 31320375*A^4*B^2*a^2*b^4 - 42595710*A^5*B*a*b^5 + 24137569*A^6*b^6)/(a^
23*b))^(1/6) - 11*(5*B*a - 17*A*b)*sqrt(x)))/((a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x
^2)*sqrt(x))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242714, size = 451, normalized size = 1.28 \[ \frac{11 \, \sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{432 \, a^{4} b} - \frac{11 \, \sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{432 \, a^{4} b} + \frac{11 \,{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{216 \, a^{4} b} + \frac{11 \,{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{216 \, a^{4} b} + \frac{11 \,{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 17 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{108 \, a^{4} b} + \frac{11 \, B a b x^{\frac{7}{2}} - 23 \, A b^{2} x^{\frac{7}{2}} + 17 \, B a^{2} \sqrt{x} - 29 \, A a b \sqrt{x}}{36 \,{\left (b x^{3} + a\right )}^{2} a^{3}} - \frac{2 \, A}{5 \, a^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/432*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*ln(sqrt(3)*sqrt(x)*(
a/b)^(1/6) + x + (a/b)^(1/3))/(a^4*b) - 11/432*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 17
*(a*b^5)^(1/6)*A*b)*ln(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^4*b) +
11/216*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6)
+ 2*sqrt(x))/(a/b)^(1/6))/(a^4*b) + 11/216*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1
/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*sqrt(x))/(a/b)^(1/6))/(a^4*b) + 11/108
*(5*(a*b^5)^(1/6)*B*a - 17*(a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a^4*b
) + 1/36*(11*B*a*b*x^(7/2) - 23*A*b^2*x^(7/2) + 17*B*a^2*sqrt(x) - 29*A*a*b*sqrt
(x))/((b*x^3 + a)^2*a^3) - 2/5*A/(a^3*x^(5/2))