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

Optimal. Leaf size=171 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{5/2}}-\frac{b \sqrt{a+b x} \sqrt{c+d x} (3 b c-7 a d)}{4 d^2}+\frac{b (a+b x)^{3/2} \sqrt{c+d x}}{2 d} \]

[Out]

-(b*(3*b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d^2) + (b*(a + b*x)^(3/2)*Sq
rt[c + d*x])/(2*d) - (2*a^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c
+ d*x])])/Sqrt[c] + (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*ArcTanh[(Sqrt
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(5/2))

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Rubi [A]  time = 0.498144, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}+\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{5/2}}-\frac{b \sqrt{a+b x} \sqrt{c+d x} (3 b c-7 a d)}{4 d^2}+\frac{b (a+b x)^{3/2} \sqrt{c+d x}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(3*b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d^2) + (b*(a + b*x)^(3/2)*Sq
rt[c + d*x])/(2*d) - (2*a^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c
+ d*x])])/Sqrt[c] + (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*ArcTanh[(Sqrt
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*d^(5/2))

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Rubi in Sympy [A]  time = 44.5587, size = 162, normalized size = 0.95 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \left (15 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{4 d^{\frac{5}{2}}} + \frac{b \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 d} + \frac{b \sqrt{a + b x} \sqrt{c + d x} \left (7 a d - 3 b c\right )}{4 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/sqrt(c) + sqrt(
b)*(15*a**2*d**2 - 10*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d
)*sqrt(a + b*x)))/(4*d**(5/2)) + b*(a + b*x)**(3/2)*sqrt(c + d*x)/(2*d) + b*sqrt
(a + b*x)*sqrt(c + d*x)*(7*a*d - 3*b*c)/(4*d**2)

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Mathematica [A]  time = 0.661139, size = 190, normalized size = 1.11 \[ \frac{a^{5/2} \log (x)}{\sqrt{c}}+\frac{1}{8} \left (-\frac{8 a^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}+\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{5/2}}+\frac{2 b \sqrt{a+b x} \sqrt{c+d x} (9 a d-3 b c+2 b d x)}{d^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 342, normalized size = 2. \[ -{\frac{1}{8\,{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 8\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{3}{d}^{2}\sqrt{bd}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}b{d}^{2}\sqrt{ac}+10\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}cd\sqrt{ac}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{2}\sqrt{ac}-4\,x{b}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-18\,abd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+6\,{b}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(8*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)+2*a*c)/x)*a^3*d^2*(b*d)^(1/2)-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b*d^2*(a*c)^(1/2)+10*ln(1/2*(2*b*d*x+
2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^2*c*d*(a*c)^(1/2
)-3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*
b^3*c^2*(a*c)^(1/2)-4*x*b^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-18
*a*b*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+6*b^2*c*(a*c)^(1/2)*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/d^2/(b*d)^(1/2)/(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)/(sqrt(d*x + c)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.72232, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(8*a^2*d^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) + (3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^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*
b^2*c + 9*a*b*d)*sqrt(b*x + a)*sqrt(d*x + c))/d^2, 1/8*(4*a^2*d^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) + (3*b^
2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sq
rt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) + 2*(2*b^2*d*x - 3*b^2*c + 9*a*b*d)*sqr
t(b*x + a)*sqrt(d*x + c))/d^2, -1/16*(16*a^2*d^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 + 15*a^2*d^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*b^2*c + 9*a*b*d)*sqrt(b*x + a)*sqrt(d*x + c))
/d^2, -1/8*(8*a^2*d^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 + 15*a^2*d^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*b^2*c + 9*a*b*d)*sqrt(b*x + a)*sqrt(d*x + c))/d^2]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{5}{2}}}{x \sqrt{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(5/2)/(x*sqrt(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.24911, size = 335, normalized size = 1.96 \[ -\frac{{\left (\frac{16 \, \sqrt{b d} a^{3} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} - 2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b d} - \frac{3 \, b^{2} c d - 7 \, a b d^{2}}{b^{2} d^{3}}\right )} + \frac{{\left (3 \, \sqrt{b d} b^{2} c^{2} - 10 \, \sqrt{b d} a b c d + 15 \, \sqrt{b d} a^{2} d^{2}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b d^{3}}\right )} b^{2}}{8 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(16*sqrt(b*d)*a^3*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b) -
2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)/(b*d) - (3*b^2*
c*d - 7*a*b*d^2)/(b^2*d^3)) + (3*sqrt(b*d)*b^2*c^2 - 10*sqrt(b*d)*a*b*c*d + 15*s
qrt(b*d)*a^2*d^2)*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^2)/(b*d^3))*b^2/abs(b)