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

Optimal. Leaf size=197 \[ -\frac{a^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{3/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{x}+\frac{3}{2} b (a+b x)^{3/2} \sqrt{c+d x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (11 a d+b c)}{4 d} \]

[Out]

(b*(b*c + 11*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d) + (3*b*(a + b*x)^(3/2)*Sqrt
[c + d*x])/2 - ((a + b*x)^(5/2)*Sqrt[c + d*x])/x - (a^(3/2)*(5*b*c + a*d)*ArcTan
h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] - (Sqrt[b]*(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^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.657553, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{a^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{c}}-\frac{\sqrt{b} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{3/2}}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{x}+\frac{3}{2} b (a+b x)^{3/2} \sqrt{c+d x}+\frac{b \sqrt{a+b x} \sqrt{c+d x} (11 a d+b c)}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

(b*(b*c + 11*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d) + (3*b*(a + b*x)^(3/2)*Sqrt
[c + d*x])/2 - ((a + b*x)^(5/2)*Sqrt[c + d*x])/x - (a^(3/2)*(5*b*c + a*d)*ArcTan
h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] - (Sqrt[b]*(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^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 70.6048, size = 182, normalized size = 0.92 \[ - \frac{a^{\frac{3}{2}} \left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \left (15 a^{2} d^{2} + 10 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 d^{\frac{3}{2}}} + \frac{3 b \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2} + \frac{b \sqrt{a + b x} \sqrt{c + d x} \left (11 a d + b c\right )}{4 d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.567418, size = 214, normalized size = 1.09 \[ \frac{1}{8} \left (\frac{4 a^{3/2} \log (x) (a d+5 b c)}{\sqrt{c}}-\frac{4 a^{3/2} (a d+5 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}+\frac{\sqrt{b} \left (15 a^2 d^2+10 a b c d-b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{3/2}}+2 \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{4 a^2}{x}+9 a b+\frac{b^2 (c+2 d x)}{d}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 504, normalized size = 2.6 \[{\frac{1}{8\,dx}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}b{d}^{2}\sqrt{ac}+10\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{2}cd\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{b}^{3}{c}^{2}\sqrt{ac}-4\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) x{a}^{3}{d}^{2}\sqrt{bd}-20\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) x{a}^{2}bcd\sqrt{bd}+4\,{x}^{2}{b}^{2}d\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+18\,xabd\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+2\,x{b}^{2}c\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-8\,{a}^{2}d\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^2,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*a^2*b*d^2*(a*c)^(1/2)+10*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*a*b^2*c
*d*(a*c)^(1/2)-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*b^3*c^2*(a*c)^(1/2)-4*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*a^3*d^2*(b*d)^(1/2)-20*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*a^2*b*c*d*(b*d)^(1/2)+4*x^
2*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*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)+2*x*b^2*c*(a*c)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-8*a^2*d*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*(b*d)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/d/(b*d)^(1/2)/x/(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((b*x + a)^(5/2)*sqrt(d*x + c)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.81857, 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^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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)**(1/2)/x**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.645449, 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)^(5/2)*sqrt(d*x + c)/x^2,x, algorithm="giac")

[Out]

sage0*x