3.607 \(\int \frac{(a+b x)^{3/2} (c+d x)^{5/2}}{x} \, dx\)

Optimal. Leaf size=304 \[ -2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{96} \sqrt{a+b x} (c+d x)^{3/2} \left (\frac{3 a^2 d}{b}+50 a c-\frac{5 b c^2}{d}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^3 d^3-17 a^2 b c d^2-55 a b^2 c^2 d+5 b^3 c^3\right )}{64 b^2 d}-\frac{\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{3/2}}+\frac{1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 d} \]

[Out]

-((5*b^3*c^3 - 55*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*b^2*d) + ((50*a*c - (5*b*c^2)/d + (3*a^2*d)/b)*Sqrt[a + b*x]*(c + d
*x)^(3/2))/96 + ((5*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*d) + ((a + b
*x)^(3/2)*(c + d*x)^(5/2))/4 - 2*a^(3/2)*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])
/(Sqrt[a]*Sqrt[c + d*x])] - ((5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 +
20*a^3*b*c*d^3 - 3*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*
x])])/(64*b^(5/2)*d^(3/2))

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Rubi [A]  time = 0.968225, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{96} \sqrt{a+b x} (c+d x)^{3/2} \left (\frac{3 a^2 d}{b}+50 a c-\frac{5 b c^2}{d}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^3 d^3-17 a^2 b c d^2-55 a b^2 c^2 d+5 b^3 c^3\right )}{64 b^2 d}-\frac{\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{5/2} d^{3/2}}+\frac{1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 d} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x,x]

[Out]

-((5*b^3*c^3 - 55*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*b^2*d) + ((50*a*c - (5*b*c^2)/d + (3*a^2*d)/b)*Sqrt[a + b*x]*(c + d
*x)^(3/2))/96 + ((5*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*d) + ((a + b
*x)^(3/2)*(c + d*x)^(5/2))/4 - 2*a^(3/2)*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])
/(Sqrt[a]*Sqrt[c + d*x])] - ((5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 +
20*a^3*b*c*d^3 - 3*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*
x])])/(64*b^(5/2)*d^(3/2))

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Rubi in Sympy [A]  time = 101., size = 284, normalized size = 0.93 \[ - 2 a^{\frac{3}{2}} c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{4} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a d + 5 b c\right )}{24 b} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - 5 b c\right ) \left (3 a d + b c\right )}{32 b d} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (- 64 a b^{2} c^{2} d + \left (a d - 5 b c\right ) \left (a d - b c\right ) \left (3 a d + b c\right )\right )}{64 b^{2} d} + \frac{\left (3 a^{4} d^{4} - 20 a^{3} b c d^{3} + 90 a^{2} b^{2} c^{2} d^{2} + 60 a b^{3} c^{3} d - 5 b^{4} c^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 b^{\frac{5}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(d*x+c)**(5/2)/x,x)

[Out]

-2*a**(3/2)*c**(5/2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x))) + (a +
b*x)**(3/2)*(c + d*x)**(5/2)/4 + (a + b*x)**(3/2)*(c + d*x)**(3/2)*(3*a*d + 5*b
*c)/(24*b) - sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - 5*b*c)*(3*a*d + b*c)/(32*b*d)
- sqrt(a + b*x)*sqrt(c + d*x)*(-64*a*b**2*c**2*d + (a*d - 5*b*c)*(a*d - b*c)*(3
*a*d + b*c))/(64*b**2*d) + (3*a**4*d**4 - 20*a**3*b*c*d**3 + 90*a**2*b**2*c**2*d
**2 + 60*a*b**3*c**3*d - 5*b**4*c**4)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(
a + b*x)))/(64*b**(5/2)*d**(3/2))

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Mathematica [A]  time = 0.238266, size = 291, normalized size = 0.96 \[ -a^{3/2} c^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+a^{3/2} c^{5/2} \log (x)+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-9 a^3 d^3+3 a^2 b d^2 (19 c+2 d x)+a b^2 d \left (337 c^2+244 c d x+72 d^2 x^2\right )+b^3 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^2 d}+\frac{\left (3 a^4 d^4-20 a^3 b c d^3+90 a^2 b^2 c^2 d^2+60 a b^3 c^3 d-5 b^4 c^4\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{5/2} d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x,x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-9*a^3*d^3 + 3*a^2*b*d^2*(19*c + 2*d*x) + a*b^2*d*
(337*c^2 + 244*c*d*x + 72*d^2*x^2) + b^3*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 +
48*d^3*x^3)))/(192*b^2*d) + a^(3/2)*c^(5/2)*Log[x] - a^(3/2)*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]] + ((-5*b^4*c^4
+ 60*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 3*a^4*d^4)*Log[b*c + a*
d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(128*b^(5/2)*d^(3/
2))

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Maple [B]  time = 0.025, size = 828, normalized size = 2.7 \[ -{\frac{1}{384\,{b}^{2}d}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-144\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-272\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+384\,{a}^{2}{c}^{3}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){b}^{2}d\sqrt{bd}-9\,{d}^{4}{a}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}+60\,{d}^{3}{a}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) c\sqrt{ac}b-270\,{c}^{2}{d}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}\sqrt{ac}{b}^{2}-180\,{b}^{3}{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a\sqrt{ac}d+15\,{b}^{4}{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}-12\,{d}^{3}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{ac}b\sqrt{bd}-488\,{d}^{2}a\sqrt{d{x}^{2}b+adx+bcx+ac}xc\sqrt{ac}{b}^{2}\sqrt{bd}-236\,{b}^{3}{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{ac}d\sqrt{bd}+18\,{d}^{3}{a}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}\sqrt{bd}-114\,{d}^{2}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}c\sqrt{ac}b\sqrt{bd}-674\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}a\sqrt{ac}{b}^{2}d\sqrt{bd}-30\,{b}^{3}{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(d*x+c)^(5/2)/x,x)

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*x^3*b^3*d^3*(b*d)^(1/2)*(a*c)^(1/2)*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)-144*x^2*a*b^2*d^3*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)-272*x^2*b^3*c*d^2*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)+384*a^2*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)*b^2*d*(b*d)^(1/2)-9*d^4*a^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*c)^(1/2)+60*d^3*a^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))*c
*(a*c)^(1/2)*b-270*c^2*d^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))*a^2*(a*c)^(1/2)*b^2-180*b^3*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))*a*(a*c)^(1/2)
*d+15*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*c)^(1/2)-12*d^3*a^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*(a*c)
^(1/2)*b*(b*d)^(1/2)-488*d^2*a*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*c*(a*c)^(1/2)*b
^2*(b*d)^(1/2)-236*b^3*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*(a*c)^(1/2)*d*(b*d)
^(1/2)+18*d^3*a^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2)*(b*d)^(1/2)-114*d^
2*a^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*c*(a*c)^(1/2)*b*(b*d)^(1/2)-674*c^2*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*a*(a*c)^(1/2)*b^2*d*(b*d)^(1/2)-30*b^3*c^3*(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)/(a
*c)^(1/2)/b^2/d/(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)^(3/2)*(d*x + c)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.5053, 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)^(3/2)*(d*x + c)^(5/2)/x,x, algorithm="fricas")

[Out]

[1/768*(384*sqrt(a*c)*sqrt(b*d)*a*b^2*c^2*d*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*(48*b^3*d^3*x^3 + 15*b^3*c^3 + 337*a*b^2*
c^2*d + 57*a^2*b*c*d^2 - 9*a^3*d^3 + 8*(17*b^3*c*d^2 + 9*a*b^2*d^3)*x^2 + 2*(59*
b^3*c^2*d + 122*a*b^2*c*d^2 + 3*a^2*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x +
c) - 3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^
4*d^4)*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), 1/384*(192*sqrt(a*c)*sqrt(-b*d)*a*b^2*c^2*d*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)*sqr
t(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 2*(48*b^3*d^3*x^3 + 1
5*b^3*c^3 + 337*a*b^2*c^2*d + 57*a^2*b*c*d^2 - 9*a^3*d^3 + 8*(17*b^3*c*d^2 + 9*a
*b^2*d^3)*x^2 + 2*(59*b^3*c^2*d + 122*a*b^2*c*d^2 + 3*a^2*b*d^3)*x)*sqrt(-b*d)*s
qrt(b*x + a)*sqrt(d*x + c) - 3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2
+ 20*a^3*b*c*d^3 - 3*a^4*d^4)*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), -1/768*(768*sqrt(-a*c)*sqrt(b*
d)*a*b^2*c^2*d*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt
(d*x + c))) - 4*(48*b^3*d^3*x^3 + 15*b^3*c^3 + 337*a*b^2*c^2*d + 57*a^2*b*c*d^2
- 9*a^3*d^3 + 8*(17*b^3*c*d^2 + 9*a*b^2*d^3)*x^2 + 2*(59*b^3*c^2*d + 122*a*b^2*c
*d^2 + 3*a^2*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(5*b^4*c^4 - 60
*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*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), -
1/384*(384*sqrt(-a*c)*sqrt(-b*d)*a*b^2*c^2*d*arctan(1/2*(2*a*c + (b*c + a*d)*x)/
(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 2*(48*b^3*d^3*x^3 + 15*b^3*c^3 + 337
*a*b^2*c^2*d + 57*a^2*b*c*d^2 - 9*a^3*d^3 + 8*(17*b^3*c*d^2 + 9*a*b^2*d^3)*x^2 +
2*(59*b^3*c^2*d + 122*a*b^2*c*d^2 + 3*a^2*b*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sq
rt(d*x + c) + 3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^
3 - 3*a^4*d^4)*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)]

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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)**(3/2)*(d*x+c)**(5/2)/x,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.370506, size = 576, normalized size = 1.89 \[ -\frac{2 \, \sqrt{b d} a^{2} c^{3}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{192} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} d^{2}{\left | b \right |}}{b^{4}} + \frac{17 \, b^{10} c d^{7}{\left | b \right |} - 9 \, a b^{9} d^{8}{\left | b \right |}}{b^{13} d^{6}}\right )} + \frac{59 \, b^{11} c^{2} d^{6}{\left | b \right |} - 14 \, a b^{10} c d^{7}{\left | b \right |} + 3 \, a^{2} b^{9} d^{8}{\left | b \right |}}{b^{13} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{12} c^{3} d^{5}{\left | b \right |} + 73 \, a b^{11} c^{2} d^{6}{\left | b \right |} - 17 \, a^{2} b^{10} c d^{7}{\left | b \right |} + 3 \, a^{3} b^{9} d^{8}{\left | b \right |}\right )}}{b^{13} d^{6}}\right )} \sqrt{b x + a} + \frac{{\left (5 \, \sqrt{b d} b^{4} c^{4}{\left | b \right |} - 60 \, \sqrt{b d} a b^{3} c^{3} d{\left | b \right |} - 90 \, \sqrt{b d} a^{2} b^{2} c^{2} d^{2}{\left | b \right |} + 20 \, \sqrt{b d} a^{3} b c d^{3}{\left | b \right |} - 3 \, \sqrt{b d} a^{4} d^{4}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, b^{4} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*(d*x + c)^(5/2)/x,x, algorithm="giac")

[Out]

-2*sqrt(b*d)*a^2*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) + 1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x
+ a)*d^2*abs(b)/b^4 + (17*b^10*c*d^7*abs(b) - 9*a*b^9*d^8*abs(b))/(b^13*d^6)) +
(59*b^11*c^2*d^6*abs(b) - 14*a*b^10*c*d^7*abs(b) + 3*a^2*b^9*d^8*abs(b))/(b^13*
d^6)) + 3*(5*b^12*c^3*d^5*abs(b) + 73*a*b^11*c^2*d^6*abs(b) - 17*a^2*b^10*c*d^7*
abs(b) + 3*a^3*b^9*d^8*abs(b))/(b^13*d^6))*sqrt(b*x + a) + 1/128*(5*sqrt(b*d)*b^
4*c^4*abs(b) - 60*sqrt(b*d)*a*b^3*c^3*d*abs(b) - 90*sqrt(b*d)*a^2*b^2*c^2*d^2*ab
s(b) + 20*sqrt(b*d)*a^3*b*c*d^3*abs(b) - 3*sqrt(b*d)*a^4*d^4*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^2)