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

Optimal. Leaf size=211 \[ \frac{\sqrt{c} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2}}+\frac{d^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{4 a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (11 a d+b c)}{4 a} \]

[Out]

(d*(b*c + 11*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*a) - ((b*c + 5*a*d)*Sqrt[a + b
*x]*(c + d*x)^(3/2))/(4*a*x) - (Sqrt[a + b*x]*(c + d*x)^(5/2))/(2*x^2) + (Sqrt[c
]*(b^2*c^2 - 10*a*b*c*d - 15*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/(4*a^(3/2)) + (d^(3/2)*(5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b]

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Rubi [A]  time = 0.663822, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt{c} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2}}+\frac{d^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{4 a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (11 a d+b c)}{4 a} \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*c + 11*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*a) - ((b*c + 5*a*d)*Sqrt[a + b
*x]*(c + d*x)^(3/2))/(4*a*x) - (Sqrt[a + b*x]*(c + d*x)^(5/2))/(2*x^2) + (Sqrt[c
]*(b^2*c^2 - 10*a*b*c*d - 15*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S
qrt[c + d*x])])/(4*a^(3/2)) + (d^(3/2)*(5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b]

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Rubi in Sympy [A]  time = 88.6344, size = 194, normalized size = 0.92 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{2 x^{2}} + \frac{d^{\frac{3}{2}} \left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b}} + \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (11 a d + b c\right )}{4 a} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + b c\right )}{4 a x} - \frac{\sqrt{c} \left (15 a^{2} d^{2} + 10 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**3,x)

[Out]

-sqrt(a + b*x)*(c + d*x)**(5/2)/(2*x**2) + d**(3/2)*(a*d + 5*b*c)*atanh(sqrt(d)*
sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/sqrt(b) + d*sqrt(a + b*x)*sqrt(c + d*x)*(
11*a*d + b*c)/(4*a) - sqrt(a + b*x)*(c + d*x)**(3/2)*(5*a*d + b*c)/(4*a*x) - sqr
t(c)*(15*a**2*d**2 + 10*a*b*c*d - b**2*c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a
)*sqrt(c + d*x)))/(4*a**(3/2))

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Mathematica [A]  time = 0.612001, size = 236, normalized size = 1.12 \[ -\frac{\sqrt{c} \log (x) \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right )}{8 a^{3/2}}+\frac{\sqrt{c} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{3/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{c (9 a d+b c)}{4 a x}-\frac{c^2}{2 x^2}+d^2\right )+\frac{d^{3/2} (a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(d^2 - c^2/(2*x^2) - (c*(b*c + 9*a*d))/(4*a*x))*Sqrt[a + b*x]*Sqrt[c + d*x] - (S
qrt[c]*(b^2*c^2 - 10*a*b*c*d - 15*a^2*d^2)*Log[x])/(8*a^(3/2)) + (Sqrt[c]*(b^2*c
^2 - 10*a*b*c*d - 15*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]])/(8*a^(3/2)) + (d^(3/2)*(5*b*c + a*d)*Log[b*c + a*d + 2
*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*Sqrt[b])

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Maple [B]  time = 0.023, size = 512, normalized size = 2.4 \[ -{\frac{1}{8\,a{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{2}\sqrt{bd}+10\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}d\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{3}\sqrt{bd}-4\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{d}^{3}\sqrt{ac}-20\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}abc{d}^{2}\sqrt{ac}-8\,{x}^{2}a{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+18\,xacd\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,xb{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+4\,a{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^3,x)

[Out]

-1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a*(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)+10*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)-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)-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))*x^2*a^2*d^3*(a*c)^(1/2)-20*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)-8*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*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)+2*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)/x^2/(a*c)^(1/2)/(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(sqrt(b*x + a)*(d*x + c)^(5/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.32573, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(5*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) - (b^2*c^2 - 10*a*b*c*d - 15*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*(4*a*d^2*x^2 - 2*a*c^2 - (b*c^2 + 9*a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/
(a*x^2), 1/16*(8*(5*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))) - (b^2*c^2 - 10*a*b*c*d - 15*
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*(4*a*d^2*x^2 - 2*a*c^2 - (b*c^2 + 9*a*c*d)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(a*x^2), 1/8*((b^2*c^2 - 10*a*b*c*d - 15*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)
)) + 2*(5*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) + 2*(4*a*d^2*x^2 - 2*a*c^2 - (b*c^2 + 9*a*c*d)*x)*
sqrt(b*x + a)*sqrt(d*x + c))/(a*x^2), 1/8*((b^2*c^2 - 10*a*b*c*d - 15*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))) + 4*(5*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*(4*a*d^2*x^2 - 2*a*c^2 -
(b*c^2 + 9*a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*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((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.626101, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x