3.720 \(\int \frac{(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=394 \[ -\frac{77 d^{7/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^{7/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{15/4} b^{9/4}}-\frac{77 d^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^3 \sqrt{d x}}{12288 a^3 b^2 \left (a+b x^2\right )}+\frac{11 d^3 \sqrt{d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac{d^3 \sqrt{d x}}{384 a b^2 \left (a+b x^2\right )^3}-\frac{d^3 \sqrt{d x}}{32 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-(d*(d*x)^(5/2))/(10*b*(a + b*x^2)^5) - (d^3*Sqrt[d*x])/(32*b^2*(a + b*x^2)^4) +
(d^3*Sqrt[d*x])/(384*a*b^2*(a + b*x^2)^3) + (11*d^3*Sqrt[d*x])/(3072*a^2*b^2*(a
+ b*x^2)^2) + (77*d^3*Sqrt[d*x])/(12288*a^3*b^2*(a + b*x^2)) - (77*d^(7/2)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(15/4)*b^
(9/4)) + (77*d^(7/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/
(8192*Sqrt[2]*a^(15/4)*b^(9/4)) - (77*d^(7/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt
[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(15/4)*b^(9/4)) + (
77*d^(7/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[d*x]])/(16384*Sqrt[2]*a^(15/4)*b^(9/4))

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Rubi [A]  time = 0.974453, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{77 d^{7/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^{7/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{15/4} b^{9/4}}-\frac{77 d^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{15/4} b^{9/4}}+\frac{77 d^3 \sqrt{d x}}{12288 a^3 b^2 \left (a+b x^2\right )}+\frac{11 d^3 \sqrt{d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac{d^3 \sqrt{d x}}{384 a b^2 \left (a+b x^2\right )^3}-\frac{d^3 \sqrt{d x}}{32 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(5/2))/(10*b*(a + b*x^2)^5) - (d^3*Sqrt[d*x])/(32*b^2*(a + b*x^2)^4) +
(d^3*Sqrt[d*x])/(384*a*b^2*(a + b*x^2)^3) + (11*d^3*Sqrt[d*x])/(3072*a^2*b^2*(a
+ b*x^2)^2) + (77*d^3*Sqrt[d*x])/(12288*a^3*b^2*(a + b*x^2)) - (77*d^(7/2)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(15/4)*b^
(9/4)) + (77*d^(7/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/
(8192*Sqrt[2]*a^(15/4)*b^(9/4)) - (77*d^(7/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt
[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(15/4)*b^(9/4)) + (
77*d^(7/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[d*x]])/(16384*Sqrt[2]*a^(15/4)*b^(9/4))

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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((d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.478005, size = 298, normalized size = 0.76 \[ \frac{d^3 \sqrt{d x} \left (-\frac{1155 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{15/4} \sqrt{x}}+\frac{1155 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{15/4} \sqrt{x}}-\frac{2310 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{15/4} \sqrt{x}}+\frac{2310 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{15/4} \sqrt{x}}+\frac{3080 \sqrt [4]{b}}{a^3 \left (a+b x^2\right )}+\frac{1760 \sqrt [4]{b}}{a^2 \left (a+b x^2\right )^2}+\frac{49152 a \sqrt [4]{b}}{\left (a+b x^2\right )^5}-\frac{64512 \sqrt [4]{b}}{\left (a+b x^2\right )^4}+\frac{1280 \sqrt [4]{b}}{a \left (a+b x^2\right )^3}\right )}{491520 b^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^3*Sqrt[d*x]*((49152*a*b^(1/4))/(a + b*x^2)^5 - (64512*b^(1/4))/(a + b*x^2)^4
+ (1280*b^(1/4))/(a*(a + b*x^2)^3) + (1760*b^(1/4))/(a^2*(a + b*x^2)^2) + (3080*
b^(1/4))/(a^3*(a + b*x^2)) - (2310*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/
a^(1/4)])/(a^(15/4)*Sqrt[x]) + (2310*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)])/(a^(15/4)*Sqrt[x]) - (1155*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(15/4)*Sqrt[x]) + (1155*Sqrt[2]*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(15/4)*Sqrt[x])))/(491520*b^(9/4))

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Maple [A]  time = 0.034, size = 339, normalized size = 0.9 \[ -{\frac{77\,{d}^{13}a}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}}\sqrt{dx}}-{\frac{231\,{d}^{11}}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{313\,{d}^{9}}{6144\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{11\,{d}^{7}b}{384\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{13}{2}}}}+{\frac{77\,{d}^{5}{b}^{2}}{12288\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{3}} \left ( dx \right ) ^{{\frac{17}{2}}}}+{\frac{77\,{d}^{3}\sqrt{2}}{32768\,{a}^{4}{b}^{2}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({1 \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,{d}^{3}\sqrt{2}}{16384\,{a}^{4}{b}^{2}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{77\,{d}^{3}\sqrt{2}}{16384\,{a}^{4}{b}^{2}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-77/4096*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(1/2)-231/2560*d^11/(b*d^2*x^2+a*d
^2)^5/b*(d*x)^(5/2)+313/6144*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(9/2)+11/384*d^7/(b
*d^2*x^2+a*d^2)^5/a^2*b*(d*x)^(13/2)+77/12288*d^5/(b*d^2*x^2+a*d^2)^5/a^3*b^2*(d
*x)^(17/2)+77/32768*d^3/a^4/b^2*(a*d^2/b)^(1/4)*2^(1/2)*ln((d*x+(a*d^2/b)^(1/4)*
(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a
*d^2/b)^(1/2)))+77/16384*d^3/a^4/b^2*(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d
^2/b)^(1/4)*(d*x)^(1/2)+1)+77/16384*d^3/a^4/b^2*(a*d^2/b)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(a*d^2/b)^(1/4)*(d*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((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29248, size = 662, normalized size = 1.68 \[ -\frac{4620 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{14}}{a^{15} b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} b^{2} \left (-\frac{d^{14}}{a^{15} b^{9}}\right )^{\frac{1}{4}}}{\sqrt{d x} d^{3} + \sqrt{a^{8} b^{4} \sqrt{-\frac{d^{14}}{a^{15} b^{9}}} + d^{7} x}}\right ) - 1155 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{14}}{a^{15} b^{9}}\right )^{\frac{1}{4}} \log \left (77 \, a^{4} b^{2} \left (-\frac{d^{14}}{a^{15} b^{9}}\right )^{\frac{1}{4}} + 77 \, \sqrt{d x} d^{3}\right ) + 1155 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{14}}{a^{15} b^{9}}\right )^{\frac{1}{4}} \log \left (-77 \, a^{4} b^{2} \left (-\frac{d^{14}}{a^{15} b^{9}}\right )^{\frac{1}{4}} + 77 \, \sqrt{d x} d^{3}\right ) - 4 \,{\left (385 \, b^{4} d^{3} x^{8} + 1760 \, a b^{3} d^{3} x^{6} + 3130 \, a^{2} b^{2} d^{3} x^{4} - 5544 \, a^{3} b d^{3} x^{2} - 1155 \, a^{4} d^{3}\right )} \sqrt{d x}}{245760 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

-1/245760*(4620*(a^3*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4
+ 5*a^7*b^3*x^2 + a^8*b^2)*(-d^14/(a^15*b^9))^(1/4)*arctan(a^4*b^2*(-d^14/(a^15*
b^9))^(1/4)/(sqrt(d*x)*d^3 + sqrt(a^8*b^4*sqrt(-d^14/(a^15*b^9)) + d^7*x))) - 11
55*(a^3*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 + 5*a^7*b^3*x
^2 + a^8*b^2)*(-d^14/(a^15*b^9))^(1/4)*log(77*a^4*b^2*(-d^14/(a^15*b^9))^(1/4) +
77*sqrt(d*x)*d^3) + 1155*(a^3*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^
6*b^4*x^4 + 5*a^7*b^3*x^2 + a^8*b^2)*(-d^14/(a^15*b^9))^(1/4)*log(-77*a^4*b^2*(-
d^14/(a^15*b^9))^(1/4) + 77*sqrt(d*x)*d^3) - 4*(385*b^4*d^3*x^8 + 1760*a*b^3*d^3
*x^6 + 3130*a^2*b^2*d^3*x^4 - 5544*a^3*b*d^3*x^2 - 1155*a^4*d^3)*sqrt(d*x))/(a^3
*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 + 5*a^7*b^3*x^2 + a^
8*b^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((d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280652, size = 467, normalized size = 1.19 \[ \frac{1}{491520} \, d^{2}{\left (\frac{2310 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{3}} + \frac{2310 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{3}} + \frac{1155 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{3}} - \frac{1155 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{3}} + \frac{8 \,{\left (385 \, \sqrt{d x} b^{4} d^{11} x^{8} + 1760 \, \sqrt{d x} a b^{3} d^{11} x^{6} + 3130 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{4} - 5544 \, \sqrt{d x} a^{3} b d^{11} x^{2} - 1155 \, \sqrt{d x} a^{4} d^{11}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{3} b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

1/491520*d^2*(2310*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^
2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^4*b^3) + 2310*sqrt(2)*(a*b^3*d^2)^
(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1
/4))/(a^4*b^3) + 1155*sqrt(2)*(a*b^3*d^2)^(1/4)*d*ln(d*x + sqrt(2)*(a*d^2/b)^(1/
4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b^3) - 1155*sqrt(2)*(a*b^3*d^2)^(1/4)*d*ln(d*
x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b^3) + 8*(385*sqrt(d
*x)*b^4*d^11*x^8 + 1760*sqrt(d*x)*a*b^3*d^11*x^6 + 3130*sqrt(d*x)*a^2*b^2*d^11*x
^4 - 5544*sqrt(d*x)*a^3*b*d^11*x^2 - 1155*sqrt(d*x)*a^4*d^11)/((b*d^2*x^2 + a*d^
2)^5*a^3*b^2))