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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.482943, 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 \[ \frac{\sqrt{d} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2}}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (7 b c-3 a d)}{4 b^2}-\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{d \sqrt{a+b x} (c+d x)^{3/2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 45.2087, size = 162, normalized size = 0.95 \[ \frac{d \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 b} - \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (3 a d - 7 b c\right )}{4 b^{2}} + \frac{\sqrt{d} \left (3 a^{2} d^{2} - 10 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{5}{2}}} - \frac{2 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.743371, size = 190, normalized size = 1.11 \[ \frac{1}{8} \left (\frac{\sqrt{d} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{5/2}}+\frac{2 d \sqrt{a+b x} \sqrt{c+d x} (-3 a d+9 b c+2 b d x)}{b^2}-\frac{8 c^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}\right )+\frac{c^{5/2} \log (x)}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.03, size = 342, normalized size = 2. \[{\frac{1}{8\,{b}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{d}^{3}\sqrt{ac}-10\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) abc{d}^{2}\sqrt{ac}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}{c}^{2}d\sqrt{ac}-8\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){b}^{2}{c}^{3}\sqrt{bd}+4\,xb{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-6\,a{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+18\,bcd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(d*x+c)^(1/2)*(b*x+a)^(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))*a^2*d^3*(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*c*d^2*(a*c)^(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))*b^2*c^2*d
*(a*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)*b
^2*c^3*(b*d)^(1/2)+4*x*b*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*a
*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+18*b*c*d*(a*c)^(1/2)*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/b^2/(b*d)^(1/2)/(a*c)^(1/
2)

_______________________________________________________________________________________

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((d*x + c)^(5/2)/(sqrt(b*x + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.75559, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.282896, size = 343, normalized size = 2.01 \[ -\frac{2 \, \sqrt{b d} c^{3}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{4} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} d^{2}{\left | b \right |}}{b^{4}} + \frac{9 \, b^{8} c d^{3}{\left | b \right |} - 5 \, a b^{7} d^{4}{\left | b \right |}}{b^{11} d^{2}}\right )} - \frac{{\left (15 \, \sqrt{b d} b^{2} c^{2}{\left | b \right |} - 10 \, \sqrt{b d} a b c d{\left | b \right |} + 3 \, \sqrt{b d} a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*c^3*abs(b)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b) +
1/4*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d^2*abs(b)/b
^4 + (9*b^8*c*d^3*abs(b) - 5*a*b^7*d^4*abs(b))/(b^11*d^2)) - 1/8*(15*sqrt(b*d)*b
^2*c^2*abs(b) - 10*sqrt(b*d)*a*b*c*d*abs(b) + 3*sqrt(b*d)*a^2*d^2*abs(b))*ln((sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/b^4