3.660 \(\int \frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=406 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{48 c^2 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{192 a c^2 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 a^2 c^2 x}-\frac{(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{5/2}}+2 b^{5/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 c x^4} \]

[Out]

((3*b^4*c^4 - 22*a*b^3*c^3*d - 128*a^2*b^2*c^2*d^2 + 22*a^3*b*c*d^3 - 3*a^4*d^4)
*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^2*x) - ((3*b^3*c^3 + 109*a*b^2*c^2*d -
19*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*a*c^2*x^2) - ((3
*b^2*c^2 + 16*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(48*c^2*x^3) -
((b*c + a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*c*x^4) - ((a + b*x)^(5/2)*(c +
d*x)^(5/2))/(5*x^5) - ((b*c + a*d)*(3*b^4*c^4 - 28*a*b^3*c^3*d + 178*a^2*b^2*c^
2*d^2 - 28*a^3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqr
t[c + d*x])])/(128*a^(5/2)*c^(5/2)) + 2*b^(5/2)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi [A]  time = 1.29522, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2+16 a b c d+3 b^2 c^2\right )}{48 c^2 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (3 a^3 d^3-19 a^2 b c d^2+109 a b^2 c^2 d+3 b^3 c^3\right )}{192 a c^2 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3-128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 a^2 c^2 x}-\frac{(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{5/2}}+2 b^{5/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 x^5}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 c x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^6,x]

[Out]

((3*b^4*c^4 - 22*a*b^3*c^3*d - 128*a^2*b^2*c^2*d^2 + 22*a^3*b*c*d^3 - 3*a^4*d^4)
*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^2*x) - ((3*b^3*c^3 + 109*a*b^2*c^2*d -
19*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*a*c^2*x^2) - ((3
*b^2*c^2 + 16*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(48*c^2*x^3) -
((b*c + a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*c*x^4) - ((a + b*x)^(5/2)*(c +
d*x)^(5/2))/(5*x^5) - ((b*c + a*d)*(3*b^4*c^4 - 28*a*b^3*c^3*d + 178*a^2*b^2*c^
2*d^2 - 28*a^3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqr
t[c + d*x])])/(128*a^(5/2)*c^(5/2)) + 2*b^(5/2)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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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+a)**(5/2)*(d*x+c)**(5/2)/x**6,x)

[Out]

Timed out

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Mathematica [A]  time = 0.339544, size = 437, normalized size = 1.08 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )-2 a^3 b c x \left (504 c^3+1448 c^2 d x+1289 c d^2 x^2+180 d^3 x^3\right )-2 a^2 b^2 c^2 x^2 \left (372 c^2+1289 c d x+1877 d^2 x^2\right )-30 a b^3 c^3 x^3 (c+12 d x)+45 b^4 c^4 x^4\right )}{1920 a^2 c^2 x^5}+\frac{\log (x) \left (3 a^5 d^5-25 a^4 b c d^4+150 a^3 b^2 c^2 d^3+150 a^2 b^3 c^3 d^2-25 a b^4 c^4 d+3 b^5 c^5\right )}{256 a^{5/2} c^{5/2}}-\frac{\left (3 a^5 d^5-25 a^4 b c d^4+150 a^3 b^2 c^2 d^3+150 a^2 b^3 c^3 d^2-25 a b^4 c^4 d+3 b^5 c^5\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{256 a^{5/2} c^{5/2}}+b^{5/2} d^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^6,x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(45*b^4*c^4*x^4 - 30*a*b^3*c^3*x^3*(c + 12*d*x) - 2
*a^2*b^2*c^2*x^2*(372*c^2 + 1289*c*d*x + 1877*d^2*x^2) - 2*a^3*b*c*x*(504*c^3 +
1448*c^2*d*x + 1289*c*d^2*x^2 + 180*d^3*x^3) - 3*a^4*(128*c^4 + 336*c^3*d*x + 24
8*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)))/(1920*a^2*c^2*x^5) + ((3*b^5*c^5 -
25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*
a^5*d^5)*Log[x])/(256*a^(5/2)*c^(5/2)) - ((3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*
b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*Log[2*a*c + b*c*
x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(256*a^(5/2)*c^(5/2)
) + b^(5/2)*d^(5/2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sq
rt[c + d*x]]

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Maple [B]  time = 0.03, size = 1146, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2*(45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a^5*d^5*(b*d)^(1/2)-375*ln((a*d*x+b*
c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a^4*b*c*d^4*(b*d
)^(1/2)+2250*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c
)/x)*x^5*a^3*b^2*c^2*d^3*(b*d)^(1/2)+2250*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2
+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a^2*b^3*c^3*d^2*(b*d)^(1/2)-375*ln((a*d*x+
b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^5*a*b^4*c^4*d*(b
*d)^(1/2)+45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c
)/x)*x^5*b^5*c^5*(b*d)^(1/2)-3840*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^5*a^2*b^3*c^2*d^3*(a*c)^(1/2)-90*a^4*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^4*x^4*(a*c)^(1/2)*(b*d)^(1/2)+720*(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)*d^3*b*a^3*c*x^4*(a*c)^(1/2)*(b*d)^(1/2)+7508*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*d^2*b^2*a^2*c^2*x^4*(a*c)^(1/2)*(b*d)^(1/2)+720*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*d*b^3*a*c^3*x^4*(a*c)^(1/2)*(b*d)^(1/2)-90*c^4*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*b^4*x^4*(a*c)^(1/2)*(b*d)^(1/2)+60*a^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*d^3*c*x^3*(a*c)^(1/2)*(b*d)^(1/2)+5156*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*b
*a^3*c^2*x^3*(a*c)^(1/2)*(b*d)^(1/2)+5156*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*b^2*
a^2*c^3*x^3*(a*c)^(1/2)*(b*d)^(1/2)+60*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^3*a
*x^3*(a*c)^(1/2)*(b*d)^(1/2)+1488*a^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*c^2*x^
2*(a*c)^(1/2)*(b*d)^(1/2)+5792*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*d*a^3*c^3*x^2*(
a*c)^(1/2)*(b*d)^(1/2)+1488*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^2*a^2*x^2*(a*c
)^(1/2)*(b*d)^(1/2)+2016*a^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*c^3*x*(a*c)^(1/2)
*(b*d)^(1/2)+2016*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*a^3*x*(a*c)^(1/2)*(b*d)^
(1/2)+768*a^4*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*(b*d)^(1/2))/(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)/x^5/(a*c)^(1/2)/(b*d)^(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 + a)^(5/2)*(d*x + c)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 18.3938, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(3840*sqrt(a*c)*sqrt(b*d)*a^2*b^2*c^2*d^2*x^5*log(8*b^2*d^2*x^2 + b^2*c^
2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d
*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 15*(3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^
3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*x^5*log(-(4*(2*a^2
*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) - (8*a^2*c^2 + (b^2*c^
2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2) - 4*(384
*a^4*c^4 - (45*b^4*c^4 - 360*a*b^3*c^3*d - 3754*a^2*b^2*c^2*d^2 - 360*a^3*b*c*d^
3 + 45*a^4*d^4)*x^4 + 2*(15*a*b^3*c^4 + 1289*a^2*b^2*c^3*d + 1289*a^3*b*c^2*d^2
+ 15*a^4*c*d^3)*x^3 + 8*(93*a^2*b^2*c^4 + 362*a^3*b*c^3*d + 93*a^4*c^2*d^2)*x^2
+ 1008*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a
*c)*a^2*c^2*x^5), 1/7680*(7680*sqrt(a*c)*sqrt(-b*d)*a^2*b^2*c^2*d^2*x^5*arctan(1
/2*(2*b*d*x + b*c + a*d)/(sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))) + 15*(3*b^5*c
^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4
+ 3*a^5*d^5)*x^5*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt
(d*x + c) - (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*
c*d)*x)*sqrt(a*c))/x^2) - 4*(384*a^4*c^4 - (45*b^4*c^4 - 360*a*b^3*c^3*d - 3754*
a^2*b^2*c^2*d^2 - 360*a^3*b*c*d^3 + 45*a^4*d^4)*x^4 + 2*(15*a*b^3*c^4 + 1289*a^2
*b^2*c^3*d + 1289*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 + 8*(93*a^2*b^2*c^4 + 362*a^
3*b*c^3*d + 93*a^4*c^2*d^2)*x^2 + 1008*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(a*c)*sqrt
(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*c^2*x^5), 1/3840*(1920*sqrt(-a*c)*sqrt(b
*d)*a^2*b^2*c^2*d^2*x^5*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2
*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2
)*x) - 15*(3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^
3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c
)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(384*a^4*c^4 - (45*b^4*c^4 - 360*a*b^3*
c^3*d - 3754*a^2*b^2*c^2*d^2 - 360*a^3*b*c*d^3 + 45*a^4*d^4)*x^4 + 2*(15*a*b^3*c
^4 + 1289*a^2*b^2*c^3*d + 1289*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 + 8*(93*a^2*b^2
*c^4 + 362*a^3*b*c^3*d + 93*a^4*c^2*d^2)*x^2 + 1008*(a^3*b*c^4 + a^4*c^3*d)*x)*s
qrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^2*c^2*x^5), 1/3840*(3840*sq
rt(-a*c)*sqrt(-b*d)*a^2*b^2*c^2*d^2*x^5*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(-
b*d)*sqrt(b*x + a)*sqrt(d*x + c))) - 15*(3*b^5*c^5 - 25*a*b^4*c^4*d + 150*a^2*b^
3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 25*a^4*b*c*d^4 + 3*a^5*d^5)*x^5*arctan(1/2*(2*
a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(384*a^4*
c^4 - (45*b^4*c^4 - 360*a*b^3*c^3*d - 3754*a^2*b^2*c^2*d^2 - 360*a^3*b*c*d^3 + 4
5*a^4*d^4)*x^4 + 2*(15*a*b^3*c^4 + 1289*a^2*b^2*c^3*d + 1289*a^3*b*c^2*d^2 + 15*
a^4*c*d^3)*x^3 + 8*(93*a^2*b^2*c^4 + 362*a^3*b*c^3*d + 93*a^4*c^2*d^2)*x^2 + 100
8*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)
*a^2*c^2*x^5)]

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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+a)**(5/2)*(d*x+c)**(5/2)/x**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.81563, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(5/2)/x^6,x, algorithm="giac")

[Out]

sage0*x