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

Optimal. Leaf size=391 \[ -2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{48 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (3 a^3 d^3+109 a^2 b c d^2-19 a b^2 c^2 d+3 b^3 c^3\right )}{192 b d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3+128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^2 d^2}+\frac{(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{5/2}}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 d} \]

[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*b^2*d^2) + ((3*b^3*c^3 - 19*a*b^2*c^2*d + 109
*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*b*d^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*d^2) + ((b*c + a
*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*d) + ((a + b*x)^(5/2)*(c + d*x)^(5/2))/5
- 2*a^(5/2)*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] +
((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[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5
/2)*d^(5/2))

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Rubi [A]  time = 1.32047, antiderivative size = 391, 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 \[ -2 a^{5/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-3 a^2 d^2-16 a b c d+3 b^2 c^2\right )}{48 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (3 a^3 d^3+109 a^2 b c d^2-19 a b^2 c^2 d+3 b^3 c^3\right )}{192 b d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-3 a^4 d^4+22 a^3 b c d^3+128 a^2 b^2 c^2 d^2-22 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^2 d^2}+\frac{(a d+b c) \left (3 a^4 d^4-28 a^3 b c d^3+178 a^2 b^2 c^2 d^2-28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{5/2}}+\frac{1}{5} (a+b x)^{5/2} (c+d x)^{5/2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{8 d} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x,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*b^2*d^2) + ((3*b^3*c^3 - 19*a*b^2*c^2*d + 109
*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*b*d^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*d^2) + ((b*c + a
*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(8*d) + ((a + b*x)^(5/2)*(c + d*x)^(5/2))/5
- 2*a^(5/2)*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] +
((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[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(5
/2)*d^(5/2))

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Rubi in Sympy [A]  time = 140.587, size = 374, normalized size = 0.96 \[ - 2 a^{\frac{5}{2}} c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}{5} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d + b c\right )}{8 b} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a^{2} d^{2} - 16 a b c d - 3 b^{2} c^{2}\right )}{48 b d} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + b c\right ) \left (3 a^{2} d^{2} - 22 a b c d + 3 b^{2} c^{2}\right )}{64 b^{2} d} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a^{4} d^{4} - 22 a^{3} b c d^{3} + 128 a^{2} b^{2} c^{2} d^{2} + 22 a b^{3} c^{3} d - 3 b^{4} c^{4}\right )}{128 b^{2} d^{2}} + \frac{\left (a d + b c\right ) \left (3 a^{4} d^{4} - 28 a^{3} b c d^{3} + 178 a^{2} b^{2} c^{2} d^{2} - 28 a b^{3} c^{3} d + 3 b^{4} c^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{128 b^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/2)*c**(5/2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x))) + (a +
b*x)**(5/2)*(c + d*x)**(5/2)/5 + (a + b*x)**(5/2)*(c + d*x)**(3/2)*(a*d + b*c)/
(8*b) - (a + b*x)**(3/2)*(c + d*x)**(3/2)*(3*a**2*d**2 - 16*a*b*c*d - 3*b**2*c**
2)/(48*b*d) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d + b*c)*(3*a**2*d**2 - 22*a*b*c
*d + 3*b**2*c**2)/(64*b**2*d) + sqrt(a + b*x)*sqrt(c + d*x)*(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)/(128*b**
2*d**2) + (a*d + b*c)*(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)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c +
d*x)))/(128*b**(5/2)*d**(5/2))

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Mathematica [A]  time = 0.329999, size = 357, normalized size = 0.91 \[ -a^{5/2} c^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+a^{5/2} c^{5/2} \log (x)+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (12 c+d x)+2 a^2 b^2 d^2 \left (1877 c^2+1289 c d x+372 d^2 x^2\right )+2 a b^3 d \left (180 c^3+1289 c^2 d x+1448 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^2}+\frac{\left (3 a^5 d^5-25 a^4 b c d^4+150 a^3 b^2 c^2 d^3+150 a^2 b^3 c^3 d^2-25 a b^4 c^4 d+3 b^5 c^5\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-45*a^4*d^4 + 30*a^3*b*d^3*(12*c + d*x) + 2*a^2*b^
2*d^2*(1877*c^2 + 1289*c*d*x + 372*d^2*x^2) + 2*a*b^3*d*(180*c^3 + 1289*c^2*d*x
+ 1448*c*d^2*x^2 + 504*d^3*x^3) + b^4*(-45*c^4 + 30*c^3*d*x + 744*c^2*d^2*x^2 +
1008*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^2*d^2) + a^(5/2)*c^(5/2)*Log[x] - a^(5/2
)*c^(5/2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d
*x]] + ((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[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a
+ b*x]*Sqrt[c + d*x]])/(256*b^(5/2)*d^(5/2))

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Maple [B]  time = 0.026, size = 1116, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 58.9237, 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,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.444189, size = 722, normalized size = 1.85 \[ -\frac{2 \, \sqrt{b d} a^{3} c^{3}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{1920} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )} d^{2}{\left | b \right |}}{b^{4}} + \frac{21 \, b^{11} c d^{9}{\left | b \right |} - 11 \, a b^{10} d^{10}{\left | b \right |}}{b^{14} d^{8}}\right )} + \frac{93 \, b^{12} c^{2} d^{8}{\left | b \right |} - 16 \, a b^{11} c d^{9}{\left | b \right |} + 3 \, a^{2} b^{10} d^{10}{\left | b \right |}}{b^{14} d^{8}}\right )} + \frac{5 \,{\left (3 \, b^{13} c^{3} d^{7}{\left | b \right |} + 109 \, a b^{12} c^{2} d^{8}{\left | b \right |} - 19 \, a^{2} b^{11} c d^{9}{\left | b \right |} + 3 \, a^{3} b^{10} d^{10}{\left | b \right |}\right )}}{b^{14} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (3 \, b^{14} c^{4} d^{6}{\left | b \right |} - 22 \, a b^{13} c^{3} d^{7}{\left | b \right |} - 128 \, a^{2} b^{12} c^{2} d^{8}{\left | b \right |} + 22 \, a^{3} b^{11} c d^{9}{\left | b \right |} - 3 \, a^{4} b^{10} d^{10}{\left | b \right |}\right )}}{b^{14} d^{8}}\right )} \sqrt{b x + a} - \frac{{\left (3 \, \sqrt{b d} b^{5} c^{5}{\left | b \right |} - 25 \, \sqrt{b d} a b^{4} c^{4} d{\left | b \right |} + 150 \, \sqrt{b d} a^{2} b^{3} c^{3} d^{2}{\left | b \right |} + 150 \, \sqrt{b d} a^{3} b^{2} c^{2} d^{3}{\left | b \right |} - 25 \, \sqrt{b d} a^{4} b c d^{4}{\left | b \right |} + 3 \, \sqrt{b d} a^{5} d^{5}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{256 \, b^{4} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*a^3*c^3*abs(b)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*
b) + 1/1920*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*
(b*x + a)*d^2*abs(b)/b^4 + (21*b^11*c*d^9*abs(b) - 11*a*b^10*d^10*abs(b))/(b^14*
d^8)) + (93*b^12*c^2*d^8*abs(b) - 16*a*b^11*c*d^9*abs(b) + 3*a^2*b^10*d^10*abs(b
))/(b^14*d^8)) + 5*(3*b^13*c^3*d^7*abs(b) + 109*a*b^12*c^2*d^8*abs(b) - 19*a^2*b
^11*c*d^9*abs(b) + 3*a^3*b^10*d^10*abs(b))/(b^14*d^8))*(b*x + a) - 15*(3*b^14*c^
4*d^6*abs(b) - 22*a*b^13*c^3*d^7*abs(b) - 128*a^2*b^12*c^2*d^8*abs(b) + 22*a^3*b
^11*c*d^9*abs(b) - 3*a^4*b^10*d^10*abs(b))/(b^14*d^8))*sqrt(b*x + a) - 1/256*(3*
sqrt(b*d)*b^5*c^5*abs(b) - 25*sqrt(b*d)*a*b^4*c^4*d*abs(b) + 150*sqrt(b*d)*a^2*b
^3*c^3*d^2*abs(b) + 150*sqrt(b*d)*a^3*b^2*c^2*d^3*abs(b) - 25*sqrt(b*d)*a^4*b*c*
d^4*abs(b) + 3*sqrt(b*d)*a^5*d^5*abs(b))*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^2)/(b^4*d^3)