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

Optimal. Leaf size=119 \[ \frac{\sqrt{c} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.286578, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{c} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.0771, size = 109, normalized size = 0.92 \[ \frac{2 d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b}} - \frac{c \sqrt{a + b x} \sqrt{c + d x}}{a x} - \frac{\sqrt{c} \left (3 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.450498, size = 172, normalized size = 1.45 \[ -\frac{\sqrt{c} \log (x) (b c-3 a d)}{2 a^{3/2}}+\frac{\sqrt{c} (b c-3 a d) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 a^{3/2}}+\frac{d^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b}}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 223, normalized size = 1.9 \[{\frac{1}{2\,ax}\sqrt{bx+a}\sqrt{dx+c} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{d}^{2}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xacd\sqrt{bd}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xb{c}^{2}\sqrt{bd}-2\,c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.693204, 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)^(3/2)/(sqrt(b*x + a)*x^2),x, algorithm="fricas")

[Out]

[1/4*(2*a*d*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*c - 3*a*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*sqrt(b*x + a)*sqrt(d*x + c)*c)/(a*x), 1/
4*(4*a*d*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*c - 3*a*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*sqrt(b*x + a)*sqrt(d*x + c)*c)/(
a*x), 1/2*(a*d*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*c - 3*a*d)*x*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(s
qrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) - 2*sqrt(b*x + a)*sqrt(d*x + c)*c)/(a*
x), 1/2*(2*a*d*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*c - 3*a*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))) - 2*sqrt(b*x + a)*sqrt(d*x
+ c)*c)/(a*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)**(3/2)/x**2/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.604665, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x