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

Optimal. Leaf size=257 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+26 a b c d+5 b^2 c^2\right )}{8 b}+\frac{\left (-a^3 d^3+15 a^2 b c d^2+45 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} \sqrt{d}}-\sqrt{a} c^{3/2} (5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\frac{4}{3} b \sqrt{a+b x} (c+d x)^{5/2}+\frac{1}{12} \sqrt{a+b x} (c+d x)^{3/2} (19 a d+5 b c) \]

[Out]

((5*b^2*c^2 + 26*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b) + ((5*b*c
+ 19*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/12 + (4*b*Sqrt[a + b*x]*(c + d*x)^(5/2
))/3 - ((a + b*x)^(3/2)*(c + d*x)^(5/2))/x - Sqrt[a]*c^(3/2)*(3*b*c + 5*a*d)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + ((5*b^3*c^3 + 45*a*b^2*c
^2*d + 15*a^2*b*c*d^2 - a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c
+ d*x])])/(8*b^(3/2)*Sqrt[d])

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Rubi [A]  time = 0.910458, antiderivative size = 257, 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 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+26 a b c d+5 b^2 c^2\right )}{8 b}+\frac{\left (-a^3 d^3+15 a^2 b c d^2+45 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} \sqrt{d}}-\sqrt{a} c^{3/2} (5 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{x}+\frac{4}{3} b \sqrt{a+b x} (c+d x)^{5/2}+\frac{1}{12} \sqrt{a+b x} (c+d x)^{3/2} (19 a d+5 b c) \]

Antiderivative was successfully verified.

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

[Out]

((5*b^2*c^2 + 26*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b) + ((5*b*c
+ 19*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/12 + (4*b*Sqrt[a + b*x]*(c + d*x)^(5/2
))/3 - ((a + b*x)^(3/2)*(c + d*x)^(5/2))/x - Sqrt[a]*c^(3/2)*(3*b*c + 5*a*d)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + ((5*b^3*c^3 + 45*a*b^2*c
^2*d + 15*a^2*b*c*d^2 - a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c
+ d*x])])/(8*b^(3/2)*Sqrt[d])

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Rubi in Sympy [A]  time = 124.949, size = 243, normalized size = 0.95 \[ - \sqrt{a} c^{\frac{3}{2}} \left (5 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{4 b \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{3} + \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{19 a d}{12} + \frac{5 b c}{12}\right ) - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{x} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a^{2} d^{2} + 26 a b c d + 5 b^{2} c^{2}\right )}{8 b} - \frac{\left (a^{3} d^{3} - 15 a^{2} b c d^{2} - 45 a b^{2} c^{2} d - 5 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{3}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*c**(3/2)*(5*a*d + 3*b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c +
d*x))) + 4*b*sqrt(a + b*x)*(c + d*x)**(5/2)/3 + sqrt(a + b*x)*(c + d*x)**(3/2)*(
19*a*d/12 + 5*b*c/12) - (a + b*x)**(3/2)*(c + d*x)**(5/2)/x + sqrt(a + b*x)*sqrt
(c + d*x)*(a**2*d**2 + 26*a*b*c*d + 5*b**2*c**2)/(8*b) - (a**3*d**3 - 15*a**2*b*
c*d**2 - 45*a*b**2*c**2*d - 5*b**3*c**3)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sq
rt(c + d*x)))/(8*b**(3/2)*sqrt(d))

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Mathematica [A]  time = 0.245942, size = 269, normalized size = 1.05 \[ \frac{1}{16} \left (\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2 x+2 a b \left (-12 c^2+34 c d x+7 d^2 x^2\right )+b^2 x \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )}{3 b x}+\frac{\left (-a^3 d^3+15 a^2 b c d^2+45 a b^2 c^2 d+5 b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2} \sqrt{d}}+8 \sqrt{a} c^{3/2} \log (x) (5 a d+3 b c)-8 \sqrt{a} c^{3/2} (5 a d+3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a + b*x]*Sqrt[c + d*x]*(3*a^2*d^2*x + 2*a*b*(-12*c^2 + 34*c*d*x + 7*d^2
*x^2) + b^2*x*(33*c^2 + 26*c*d*x + 8*d^2*x^2)))/(3*b*x) + 8*Sqrt[a]*c^(3/2)*(3*b
*c + 5*a*d)*Log[x] - 8*Sqrt[a]*c^(3/2)*(3*b*c + 5*a*d)*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^3*c^3 + 45*a*b^2*c^2*d
+ 15*a^2*b*c*d^2 - a^3*d^3)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a
+ b*x]*Sqrt[c + d*x]])/(b^(3/2)*Sqrt[d]))/16

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Maple [B]  time = 0.025, size = 696, normalized size = 2.7 \[ -{\frac{1}{48\,bx}\sqrt{bx+a}\sqrt{dx+c} \left ( -16\,{x}^{3}{b}^{2}{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+3\,{d}^{3}{a}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x\sqrt{ac}-45\,{d}^{2}{a}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cbx\sqrt{ac}-135\,da{b}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}x\sqrt{ac}-15\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{3}x\sqrt{ac}+120\,{a}^{2}{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) db\sqrt{bd}x+72\,a{c}^{3}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){b}^{2}\sqrt{bd}x-28\,{x}^{2}a{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}\sqrt{ac}-52\,{x}^{2}{b}^{2}cd\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-6\,{d}^{2}{a}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}x\sqrt{ac}-136\,acd\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}x\sqrt{ac}-66\,{b}^{2}{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}x\sqrt{ac}+48\,ab{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-16*x^3*b^2*d^2*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)+3*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))*x*(a*c)^(1/2)-45*d^2*a^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))*c*b*x*(a*c
)^(1/2)-135*d*a*b^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))*c^2*x*(a*c)^(1/2)-15*b^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^3*x*(a*c)^(1/2)+120*a^2*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*b*(
b*d)^(1/2)*x+72*a*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*(b*d)^(1/2)*x-28*x^2*a*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(
b*d)^(1/2)*(a*c)^(1/2)-52*x^2*b^2*c*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)*(a*c)^(1/2)-6*d^2*a^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*x*(a*c)^(1/2
)-136*a*c*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*d)^(1/2)*x*(a*c)^(1/2)-66*b^2*c
^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*x*(a*c)^(1/2)+48*a*b*c^2*(b*d)^(1
/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
/b/(b*d)^(1/2)/x/(a*c)^(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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.75, 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^2,x, algorithm="fricas")

[Out]

[1/96*(24*(3*b^2*c^2 + 5*a*b*c*d)*sqrt(a*c)*sqrt(b*d)*x*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) - 3*(5*b^3*c^3 + 45*a*b^2*c^2*d +
15*a^2*b*c*d^2 - a^3*d^3)*x*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)) + 4*(8*b^2*d^2*x^3 - 24*a*b*c^2 + 2*(13*b^2*c*d + 7*a*b
*d^2)*x^2 + (33*b^2*c^2 + 68*a*b*c*d + 3*a^2*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqr
t(d*x + c))/(sqrt(b*d)*b*x), 1/48*(12*(3*b^2*c^2 + 5*a*b*c*d)*sqrt(a*c)*sqrt(-b*
d)*x*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) + 3*
(5*b^3*c^3 + 45*a*b^2*c^2*d + 15*a^2*b*c*d^2 - a^3*d^3)*x*arctan(1/2*(2*b*d*x +
b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)) + 2*(8*b^2*d^2*x^3 - 24
*a*b*c^2 + 2*(13*b^2*c*d + 7*a*b*d^2)*x^2 + (33*b^2*c^2 + 68*a*b*c*d + 3*a^2*d^2
)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*b*x), -1/96*(48*(3*b^2*
c^2 + 5*a*b*c*d)*sqrt(-a*c)*sqrt(b*d)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt
(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) + 3*(5*b^3*c^3 + 45*a*b^2*c^2*d + 15*a^2*b*
c*d^2 - a^3*d^3)*x*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)) - 4*(8*b^2*d^2*x^3 - 24*a*b*c^2 + 2*(13*b^2*c*d + 7*a*b*d^2)*x^2
+ (33*b^2*c^2 + 68*a*b*c*d + 3*a^2*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c)
)/(sqrt(b*d)*b*x), -1/48*(24*(3*b^2*c^2 + 5*a*b*c*d)*sqrt(-a*c)*sqrt(-b*d)*x*arc
tan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 3*(5
*b^3*c^3 + 45*a*b^2*c^2*d + 15*a^2*b*c*d^2 - a^3*d^3)*x*arctan(1/2*(2*b*d*x + b*
c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)) - 2*(8*b^2*d^2*x^3 - 24*a
*b*c^2 + 2*(13*b^2*c*d + 7*a*b*d^2)*x^2 + (33*b^2*c^2 + 68*a*b*c*d + 3*a^2*d^2)*
x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*b*x)]

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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**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.65058, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x