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

Optimal. Leaf size=258 \[ -\frac{3 \sqrt{c} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a}}+\frac{3 \sqrt{d} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b}}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{4 c x}+\frac{d \sqrt{a+b x} (c+d x)^{3/2} (5 a d+7 b c)}{4 c}+3 d \sqrt{a+b x} \sqrt{c+d x} (a d+b c) \]

[Out]

3*d*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x] + (d*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(
c + d*x)^(3/2))/(4*c) - ((3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(4*c*x)
- ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(2*x^2) - (3*Sqrt[c]*(b^2*c^2 + 10*a*b*c*d +
5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*Sqrt[a]
) + (3*Sqrt[d]*(5*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x]
)/(Sqrt[b]*Sqrt[c + d*x])])/(4*Sqrt[b])

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Rubi [A]  time = 0.878047, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{3 \sqrt{c} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a}}+\frac{3 \sqrt{d} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b}}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{4 c x}+\frac{d \sqrt{a+b x} (c+d x)^{3/2} (5 a d+7 b c)}{4 c}+3 d \sqrt{a+b x} \sqrt{c+d x} (a d+b c) \]

Antiderivative was successfully verified.

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

[Out]

3*d*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x] + (d*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(
c + d*x)^(3/2))/(4*c) - ((3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(4*c*x)
- ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(2*x^2) - (3*Sqrt[c]*(b^2*c^2 + 10*a*b*c*d +
5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(4*Sqrt[a]
) + (3*Sqrt[d]*(5*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x]
)/(Sqrt[b]*Sqrt[c + d*x])])/(4*Sqrt[b])

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Rubi in Sympy [A]  time = 123.838, size = 245, normalized size = 0.95 \[ 3 d \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right ) - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{2 x^{2}} + \frac{d \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + 7 b c\right )}{4 c} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (5 a d + 3 b c\right )}{4 c x} + \frac{3 \sqrt{d} \left (a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{4 \sqrt{b}} - \frac{3 \sqrt{c} \left (5 a^{2} d^{2} + 10 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*d*sqrt(a + b*x)*sqrt(c + d*x)*(a*d + b*c) - (a + b*x)**(3/2)*(c + d*x)**(5/2)/
(2*x**2) + d*sqrt(a + b*x)*(c + d*x)**(3/2)*(5*a*d + 7*b*c)/(4*c) - sqrt(a + b*x
)*(c + d*x)**(5/2)*(5*a*d + 3*b*c)/(4*c*x) + 3*sqrt(d)*(a**2*d**2 + 10*a*b*c*d +
5*b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x)))/(4*sqrt(b)) -
3*sqrt(c)*(5*a**2*d**2 + 10*a*b*c*d + b**2*c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(s
qrt(a)*sqrt(c + d*x)))/(4*sqrt(a))

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Mathematica [A]  time = 0.220932, size = 263, normalized size = 1.02 \[ \frac{1}{8} \left (\frac{3 \sqrt{c} \log (x) \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )}{\sqrt{a}}-\frac{3 \sqrt{c} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+\frac{3 \sqrt{d} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (a \left (-2 c^2-9 c d x+5 d^2 x^2\right )+b x \left (-5 c^2+9 c d x+2 d^2 x^2\right )\right )}{x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a + b*x]*Sqrt[c + d*x]*(b*x*(-5*c^2 + 9*c*d*x + 2*d^2*x^2) + a*(-2*c^2
- 9*c*d*x + 5*d^2*x^2)))/x^2 + (3*Sqrt[c]*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Log
[x])/Sqrt[a] - (3*Sqrt[c]*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Log[2*a*c + b*c*x +
a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/Sqrt[a] + (3*Sqrt[d]*(5
*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqr
t[a + b*x]*Sqrt[c + d*x]])/Sqrt[b])/8

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Maple [B]  time = 0.024, size = 650, normalized size = 2.5 \[ -{\frac{1}{8\,{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{2}\sqrt{bd}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}d\sqrt{bd}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{3}\sqrt{bd}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{d}^{3}\sqrt{ac}-30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}abc{d}^{2}\sqrt{ac}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}{c}^{2}d\sqrt{ac}-4\,{x}^{3}b{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-10\,{x}^{2}a{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-18\,{x}^{2}bcd\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+18\,xacd\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+10\,xb{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+4\,a{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(15*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^2*a^2*c*d^2*(b*d)^(1/2)+30*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^2*a*b*c^2*d*(b*d)^(1/2)+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)*x^2*b^2*c^
3*(b*d)^(1/2)-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^2*a^2*d^3*(a*c)^(1/2)-30*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^2*a*b*c*d^2*(a*c)^(1/2)-15*
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^2*b^2*c^2*d*(a*c)^(1/2)-4*x^3*b*d^2*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)-10*x^2*a*d^2*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)-18*x^2*b*c*d*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+18*x
*a*c*d*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+10*x*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)+4*a*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)/(a*c)^(1/2)/(
b*d)^(1/2)/x^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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.14768, 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^3,x, algorithm="fricas")

[Out]

[1/16*(3*(5*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^2*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^
2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d
*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + 3*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d
^2)*x^2*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^
2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^
2*c*d)*x)/x^2) + 4*(2*b*d^2*x^3 - 2*a*c^2 + (9*b*c*d + 5*a*d^2)*x^2 - (5*b*c^2 +
9*a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/x^2, 1/16*(6*(5*b^2*c^2 + 10*a*b*c*d +
a^2*d^2)*x^2*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*
x + c)*b*sqrt(-d/b))) + 3*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*x^2*sqrt(c/a)*log((
8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x
)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(2*b
*d^2*x^3 - 2*a*c^2 + (9*b*c*d + 5*a*d^2)*x^2 - (5*b*c^2 + 9*a*c*d)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/x^2, -1/16*(6*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*x^2*sqrt(-c/
a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))
) - 3*(5*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^2*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c
^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x
+ c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(2*b*d^2*x^3 - 2*a*c^2 + (9*b*c*d
+ 5*a*d^2)*x^2 - (5*b*c^2 + 9*a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/x^2, -1/8*(
3*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*x^2*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a
*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) - 3*(5*b^2*c^2 + 10*a*b*c*d +
a^2*d^2)*x^2*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*
x + c)*b*sqrt(-d/b))) - 2*(2*b*d^2*x^3 - 2*a*c^2 + (9*b*c*d + 5*a*d^2)*x^2 - (5*
b*c^2 + 9*a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/x^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)**(3/2)*(d*x+c)**(5/2)/x**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.687555, 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^3,x, algorithm="giac")

[Out]

sage0*x