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

Optimal. Leaf size=299 \[ -2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (64 a^2 b c d^2+(b c-5 a d) (b c-a d) (a d+3 b c)\right )}{64 b d^2}+\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-5 a d) (a d+3 b c)}{32 d^2}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 d} \]

[Out]

((64*a^2*b*c*d^2 + (b*c - 5*a*d)*(b*c - a*d)*(3*b*c + a*d))*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*b*d^2) - ((b*c - 5*a*d)*(3*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)
)/(32*d^2) + ((3*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*d) + ((a + b*
x)^(5/2)*(c + d*x)^(3/2))/4 - 2*a^(5/2)*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])] + ((3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 6
0*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x
])])/(64*b^(3/2)*d^(5/2))

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Rubi [A]  time = 1.11111, antiderivative size = 294, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -2 a^{5/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{64} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{5 a^3 d}{b}+73 a^2 c-\frac{17 a b c^2}{d}+\frac{3 b^2 c^3}{d^2}\right )+\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-5 a d) (a d+3 b c)}{32 d^2}+\frac{1}{4} (a+b x)^{5/2} (c+d x)^{3/2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (5 a d+3 b c)}{24 d} \]

Antiderivative was successfully verified.

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

[Out]

((73*a^2*c + (3*b^2*c^3)/d^2 - (17*a*b*c^2)/d + (5*a^3*d)/b)*Sqrt[a + b*x]*Sqrt[
c + d*x])/64 - ((b*c - 5*a*d)*(3*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(32*d
^2) + ((3*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*d) + ((a + b*x)^(5/2
)*(c + d*x)^(3/2))/4 - 2*a^(5/2)*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a
]*Sqrt[c + d*x])] + ((3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b
*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(6
4*b^(3/2)*d^(5/2))

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Rubi in Sympy [A]  time = 98.3013, size = 298, normalized size = 1. \[ - 2 a^{\frac{5}{2}} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{4} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (5 a d + 3 b c\right )}{24 b} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (5 a^{2} d^{2} - 50 a b c d - 3 b^{2} c^{2}\right )}{96 b d} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (5 a^{3} d^{3} - 55 a^{2} b c d^{2} - 17 a b^{2} c^{2} d + 3 b^{3} c^{3}\right )}{64 b d^{2}} - \frac{\left (5 a^{4} d^{4} - 60 a^{3} b c d^{3} - 90 a^{2} b^{2} c^{2} d^{2} + 20 a b^{3} c^{3} d - 3 b^{4} c^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/2)*c**(3/2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x))) + (a +
b*x)**(5/2)*(c + d*x)**(3/2)/4 + (a + b*x)**(5/2)*sqrt(c + d*x)*(5*a*d + 3*b*c)
/(24*b) - (a + b*x)**(3/2)*sqrt(c + d*x)*(5*a**2*d**2 - 50*a*b*c*d - 3*b**2*c**2
)/(96*b*d) - sqrt(a + b*x)*sqrt(c + d*x)*(5*a**3*d**3 - 55*a**2*b*c*d**2 - 17*a*
b**2*c**2*d + 3*b**3*c**3)/(64*b*d**2) - (5*a**4*d**4 - 60*a**3*b*c*d**3 - 90*a*
*2*b**2*c**2*d**2 + 20*a*b**3*c**3*d - 3*b**4*c**4)*atanh(sqrt(d)*sqrt(a + b*x)/
(sqrt(b)*sqrt(c + d*x)))/(64*b**(3/2)*d**(5/2))

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Mathematica [A]  time = 0.211053, size = 290, normalized size = 0.97 \[ -a^{5/2} c^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+a^{5/2} c^{3/2} \log (x)+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^3 d^3+a^2 b d^2 (337 c+118 d x)+a b^2 d \left (57 c^2+244 c d x+136 d^2 x^2\right )+b^3 \left (-9 c^3+6 c^2 d x+72 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d^2}+\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{3/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^3*d^3 + a^2*b*d^2*(337*c + 118*d*x) + a*b^2*d
*(57*c^2 + 244*c*d*x + 136*d^2*x^2) + b^3*(-9*c^3 + 6*c^2*d*x + 72*c*d^2*x^2 + 4
8*d^3*x^3)))/(192*b*d^2) + a^(5/2)*c^(3/2)*Log[x] - a^(5/2)*c^(3/2)*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^4*c^4 - 2
0*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*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^(3/2)*d^(5/2))

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Maple [B]  time = 0.023, size = 828, normalized size = 2.8 \[ -{\frac{1}{384\,{d}^{2}b}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-272\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-144\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+384\,{a}^{3}{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){d}^{2}b\sqrt{bd}+15\,{d}^{4}{a}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}-180\,{d}^{3}{a}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) c\sqrt{ac}b-270\,{c}^{2}{d}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}\sqrt{ac}{b}^{2}+60\,{b}^{3}{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a\sqrt{ac}d-9\,{b}^{4}{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}-236\,{d}^{3}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{ac}b\sqrt{bd}-488\,{d}^{2}a\sqrt{d{x}^{2}b+adx+bcx+ac}xc\sqrt{ac}{b}^{2}\sqrt{bd}-12\,{b}^{3}{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}x\sqrt{ac}d\sqrt{bd}-30\,{d}^{3}{a}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}\sqrt{bd}-674\,{d}^{2}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}c\sqrt{ac}b\sqrt{bd}-114\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}a\sqrt{ac}{b}^{2}d\sqrt{bd}+18\,{b}^{3}{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(3/2)/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)-272*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)-144*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^3*c^2*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*(b*d)^(1/2)+15*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)-180*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+60*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-9*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)-236*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)-12*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)-30*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)-674*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)-114*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)+18*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)/d^2/b/(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)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.3131, 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)^(3/2)/x,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.358183, size = 574, normalized size = 1.92 \[ -\frac{2 \, \sqrt{b d} a^{3} c^{2}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{192} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )} d{\left | b \right |}}{b^{3}} + \frac{9 \, b^{6} c d^{6}{\left | b \right |} - a b^{5} d^{7}{\left | b \right |}}{b^{8} d^{6}}\right )} + \frac{3 \, b^{7} c^{2} d^{5}{\left | b \right |} + 50 \, a b^{6} c d^{6}{\left | b \right |} - 5 \, a^{2} b^{5} d^{7}{\left | b \right |}}{b^{8} d^{6}}\right )} - \frac{3 \,{\left (3 \, b^{8} c^{3} d^{4}{\left | b \right |} - 17 \, a b^{7} c^{2} d^{5}{\left | b \right |} - 55 \, a^{2} b^{6} c d^{6}{\left | b \right |} + 5 \, a^{3} b^{5} d^{7}{\left | b \right |}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} - \frac{{\left (3 \, \sqrt{b d} b^{4} c^{4}{\left | b \right |} - 20 \, \sqrt{b d} a b^{3} c^{3} d{\left | b \right |} + 90 \, \sqrt{b d} a^{2} b^{2} c^{2} d^{2}{\left | b \right |} + 60 \, \sqrt{b d} a^{3} b c d^{3}{\left | b \right |} - 5 \, \sqrt{b d} a^{4} d^{4}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*a^3*c^2*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*abs(b)/b^3 + (9*b^6*c*d^6*abs(b) - a*b^5*d^7*abs(b))/(b^8*d^6)) + (3*b^7
*c^2*d^5*abs(b) + 50*a*b^6*c*d^6*abs(b) - 5*a^2*b^5*d^7*abs(b))/(b^8*d^6)) - 3*(
3*b^8*c^3*d^4*abs(b) - 17*a*b^7*c^2*d^5*abs(b) - 55*a^2*b^6*c*d^6*abs(b) + 5*a^3
*b^5*d^7*abs(b))/(b^8*d^6))*sqrt(b*x + a) - 1/128*(3*sqrt(b*d)*b^4*c^4*abs(b) -
20*sqrt(b*d)*a*b^3*c^3*d*abs(b) + 90*sqrt(b*d)*a^2*b^2*c^2*d^2*abs(b) + 60*sqrt(
b*d)*a^3*b*c*d^3*abs(b) - 5*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^3*d^3)