3.759 \(\int \frac{(c+d x)^{5/2}}{x (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=163 \[ -\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}-\frac{d \sqrt{a+b x} \sqrt{c+d x} (2 b c-3 a d)}{a b^2}+\frac{2 (c+d x)^{3/2} (b c-a d)}{a b \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.499517, antiderivative size = 163, 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 \[ -\frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2}}+\frac{d^{3/2} (5 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}-\frac{d \sqrt{a+b x} \sqrt{c+d x} (2 b c-3 a d)}{a b^2}+\frac{2 (c+d x)^{3/2} (b c-a d)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x*(a + b*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 50.1173, size = 151, normalized size = 0.93 \[ - \frac{d^{\frac{3}{2}} \left (3 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{5}{2}}} - \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{a b \sqrt{a + b x}} + \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (3 a d - 2 b c\right )}{a b^{2}} - \frac{2 c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x/(b*x+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.447542, size = 184, normalized size = 1.13 \[ -\frac{c^{5/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2}}+\frac{c^{5/2} \log (x)}{a^{3/2}}+\frac{d^{3/2} (5 b c-3 a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 (b c-a d)^2}{a (a+b x)}+d^2\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x*(a + b*x)^(3/2)),x]

[Out]

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

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Maple [B]  time = 0.035, size = 492, normalized size = 3. \[ -{\frac{1}{2\,{b}^{2}a}\sqrt{dx+c} \left ( 2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{b}^{3}{c}^{3}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}b{d}^{3}\sqrt{ac}-5\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{2}c{d}^{2}\sqrt{ac}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) a{b}^{2}{c}^{3}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}\sqrt{ac}-5\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bc{d}^{2}\sqrt{ac}-2\,xab{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-6\,{a}^{2}{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+8\,abcd\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-4\,{b}^{2}{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x/(b*x+a)^(3/2),x)

[Out]

-1/2*(d*x+c)^(1/2)*(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^3*c^3*(b*d)^(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))*x*a^2*b*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^2*c*d^2*(a*c)^(1/2)+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)*a*b^2*c^3*(b*d)^(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^3*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))*a^2*b*c*d^2*(a*c)^(1/2)-2*x*a*b*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)*(b*d)^(1/2)-6*a^2*d^2*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
8*a*b*c*d*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-4*b^2*c^2*(b*d)^(1/2)*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/a/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(a*c)
^(1/2)/(b*x+a)^(1/2)/b^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)^(5/2)/((b*x + a)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.12519, 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)/((b*x + a)^(3/2)*x),x, algorithm="fricas")

[Out]

[-1/4*((5*a^2*b*c*d - 3*a^3*d^2 + (5*a*b^2*c*d - 3*a^2*b*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)*sqr
t(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) - 2*(b^3*c^2*x + a
*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) - 4*(a*b*d^2*x + 2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*sqrt(b*x +
a)*sqrt(d*x + c))/(a*b^3*x + a^2*b^2), 1/2*((5*a^2*b*c*d - 3*a^3*d^2 + (5*a*b^2*
c*d - 3*a^2*b*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^3*c^2*x + a*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) + 2*(a*b*d^2*x +
2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(a*b^3*x + a^2*b
^2), -1/4*(4*(b^3*c^2*x + a*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))) + (5*a^2*b*c*d - 3*a^3*d^2 + (5*a
*b^2*c*d - 3*a^2*b*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*b*d^2*x + 2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*sqrt
(b*x + a)*sqrt(d*x + c))/(a*b^3*x + a^2*b^2), -1/2*(2*(b^3*c^2*x + a*b^2*c^2)*sq
rt(-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^2*b*c*d - 3*a^3*d^2 + (5*a*b^2*c*d - 3*a^2*b*d^2)*x)*sqrt(-d/b)*a
rctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) - 2*
(a*b*d^2*x + 2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(a*
b^3*x + a^2*b^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x\right )^{\frac{5}{2}}}{x \left (a + b x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(5/2)/x/(b*x+a)**(3/2),x)

[Out]

Integral((c + d*x)**(5/2)/(x*(a + b*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.603002, 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)/((b*x + a)^(3/2)*x),x, algorithm="giac")

[Out]

sage0*x