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

Optimal. Leaf size=160 \[ \frac{c^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{a b} \]

[Out]

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

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Rubi [A]  time = 0.511261, antiderivative size = 160, 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 \[ \frac{c^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{a b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 51.506, size = 144, normalized size = 0.9 \[ - \frac{d^{\frac{3}{2}} \left (a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}}} - \frac{c \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{a x} + \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right )}{a b} - \frac{c^{\frac{3}{2}} \left (5 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.44523, size = 192, normalized size = 1.2 \[ \frac{1}{2} \left (\frac{c^{3/2} \log (x) (5 a d-b c)}{a^{3/2}}+\frac{c^{3/2} (b c-5 a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2}}+2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{d^2}{b}-\frac{c^2}{a x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 320, normalized size = 2. \[ -{\frac{1}{2\,axb}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{a}^{2}{d}^{3}\sqrt{ac}-5\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabc{d}^{2}\sqrt{ac}+5\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{2}d\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{b}^{2}{c}^{3}\sqrt{bd}-2\,xa{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+2\,b{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a*(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))*x*a^2*d^3*(a*c)^(1/2)-5*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))*x*a*b*c*d^2*(a*c)^(1/2)+5*ln
((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*a*b*c^2*d*(b*d)^
(1/2)-ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x*b^2*c^3*
(b*d)^(1/2)-2*x*a*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+2*b*c^2*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x/(b*d)^(
1/2)/(a*c)^(1/2)/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.99564, 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^2),x, algorithm="fricas")

[Out]

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

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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)/x**2/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.62561, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x