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

Optimal. Leaf size=275 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{4 c}-\frac{3 \sqrt{a} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{c}}+\frac{3 \sqrt{b} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{d}}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{4 c x}+\frac{3 b \sqrt{a+b x} (c+d x)^{3/2} (a d+3 b c)}{4 c} \]

[Out]

(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c) + (3*b*(3*
b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(4*c) - ((5*b*c + 3*a*d)*(a + b*x)^(3/
2)*(c + d*x)^(3/2))/(4*c*x) - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(2*x^2) - (3*Sqr
t[a]*(5*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]
*Sqrt[c + d*x])])/(4*Sqrt[c]) + (3*Sqrt[b]*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Ar
cTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*Sqrt[d])

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Rubi [A]  time = 0.905715, antiderivative size = 275, 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 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{4 c}-\frac{3 \sqrt{a} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{c}}+\frac{3 \sqrt{b} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{d}}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{4 c x}+\frac{3 b \sqrt{a+b x} (c+d x)^{3/2} (a d+3 b c)}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*c) + (3*b*(3*
b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(4*c) - ((5*b*c + 3*a*d)*(a + b*x)^(3/
2)*(c + d*x)^(3/2))/(4*c*x) - ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(2*x^2) - (3*Sqr
t[a]*(5*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]
*Sqrt[c + d*x])])/(4*Sqrt[c]) + (3*Sqrt[b]*(b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Ar
cTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*Sqrt[d])

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Rubi in Sympy [A]  time = 140.032, size = 245, normalized size = 0.89 \[ - \frac{3 \sqrt{a} \left (a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{c}} + \frac{3 \sqrt{b} \left (5 a^{2} d^{2} + 10 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 \sqrt{d}} + 3 b \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right ) - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{2 x^{2}} + \frac{3 d \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + 3 b c\right )}{4 c} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a d + 5 b c\right )}{4 c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.214794, size = 259, normalized size = 0.94 \[ \frac{1}{8} \left (\frac{3 \sqrt{a} \log (x) \left (a^2 d^2+10 a b c d+5 b^2 c^2\right )}{\sqrt{c}}-\frac{3 \sqrt{a} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}+\frac{3 \sqrt{b} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{d}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 (2 c+5 d x)+9 a b x (c-d x)-b^2 x^2 (5 c+2 d x)\right )}{x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 650, normalized size = 2.4 \[{\frac{1}{8\,{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}b{d}^{2}\sqrt{ac}+30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{2}cd\sqrt{ac}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{3}{c}^{2}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{3}{d}^{2}\sqrt{bd}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}bcd\sqrt{bd}-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}\sqrt{bd}+4\,{x}^{3}{b}^{2}d\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+18\,{x}^{2}abd\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+10\,{x}^{2}{b}^{2}c\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-10\,x{a}^{2}d\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-18\,xabc\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-4\,{a}^{2}c\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x+a)^(1/2)*(d*x+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*a^2*b*d^2*(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
^2*c*d*(a*c)^(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*b^3*c^2*(a*c)^(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*a^3*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^2*b*c*d*(b*d
)^(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*b^2*c^2*(b*d)^(1/2)+4*x^3*b^2*d*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*(b*d)^(1/2)+18*x^2*a*b*d*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^
(1/2)+10*x^2*b^2*c*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-10*x*
a^2*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*b*c*(b*d)^(
1/2)*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)-4*a^2*c*(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*d)^(1/2)/x^
2/(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)^(5/2)*(d*x + c)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.94736, 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^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.706793, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x