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

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

[Out]

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

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Rubi [A]  time = 0.239707, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 (b c-5 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{7/2} \sqrt{d}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-5 a d)}{4 b^3}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-5 a d)}{2 b^2 (b c-a d)}+\frac{2 a (c+d x)^{5/2}}{b \sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.1372, size = 155, normalized size = 0.91 \[ - \frac{2 a \left (c + d x\right )^{\frac{5}{2}}}{b \sqrt{a + b x} \left (a d - b c\right )} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d - b c\right )}{2 b^{2} \left (a d - b c\right )} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (5 a d - b c\right )}{4 b^{3}} + \frac{3 \left (a d - b c\right ) \left (5 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{4 b^{\frac{7}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.134885, size = 130, normalized size = 0.76 \[ \frac{\sqrt{c+d x} \left (-15 a^2 d+a b (13 c-5 d x)+b^2 x (5 c+2 d x)\right )}{4 b^3 \sqrt{a+b x}}+\frac{3 (b c-5 a d) (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{7/2} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(-15*a^2*d + a*b*(13*c - 5*d*x) + b^2*x*(5*c + 2*d*x)))/(4*b^3*Sq
rt[a + b*x]) + (3*(b*c - 5*a*d)*(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]])/(8*b^(7/2)*Sqrt[d])

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Maple [B]  time = 0.03, size = 455, normalized size = 2.7 \[{\frac{1}{8\,{b}^{3}}\sqrt{dx+c} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}b{d}^{2}-18\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xa{b}^{2}cd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{3}{c}^{2}+4\,{x}^{2}{b}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{2}-18\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}bcd+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{2}{c}^{2}-10\,xabd\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+10\,x{b}^{2}c\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-30\,{a}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+26\,abc\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(d*x+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))*x*a^2*b*d^2-18*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+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*b^3*c^2+4*x^2*b^2*d*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(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))*a^3*d^2-18*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+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*b^2*c^2-10*x*a*b*d*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)+10*x*b^2*c*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*a^2*d*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+26*a*b*c*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+
c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(1/2)/b^3

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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)*x/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.407888, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b^{2} d x^{2} + 13 \, a b c - 15 \, a^{2} d + 5 \,{\left (b^{2} c - a b d\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} +{\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{16 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b d}}, \frac{2 \,{\left (2 \, b^{2} d x^{2} + 13 \, a b c - 15 \, a^{2} d + 5 \,{\left (b^{2} c - a b d\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a b^{2} c^{2} - 6 \, a^{2} b c d + 5 \, a^{3} d^{2} +{\left (b^{3} c^{2} - 6 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{8 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*b^2*d*x^2 + 13*a*b*c - 15*a^2*d + 5*(b^2*c - a*b*d)*x)*sqrt(b*d)*sqr
t(b*x + a)*sqrt(d*x + c) + 3*(a*b^2*c^2 - 6*a^2*b*c*d + 5*a^3*d^2 + (b^3*c^2 - 6
*a*b^2*c*d + 5*a^2*b*d^2)*x)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x +
a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d +
a*b*d^2)*x)*sqrt(b*d)))/((b^4*x + a*b^3)*sqrt(b*d)), 1/8*(2*(2*b^2*d*x^2 + 13*a
*b*c - 15*a^2*d + 5*(b^2*c - a*b*d)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) +
3*(a*b^2*c^2 - 6*a^2*b*c*d + 5*a^3*d^2 + (b^3*c^2 - 6*a*b^2*c*d + 5*a^2*b*d^2)*x
)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d))
)/((b^4*x + a*b^3)*sqrt(-b*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.586243, 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)^(3/2)*x/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

sage0*x