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

Optimal. Leaf size=165 \[ \frac{2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{(5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d)}{d^3 (b c-a d)}-\frac{4 c (a+b x)^{3/2}}{d^2 \sqrt{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.411158, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 c^2 (a+b x)^{3/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac{(5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d)}{d^3 (b c-a d)}-\frac{4 c (a+b x)^{3/2}}{d^2 \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.2619, size = 146, normalized size = 0.88 \[ - \frac{2 c^{2} \left (a + b x\right )^{\frac{3}{2}}}{3 d^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 c \left (a + b x\right )^{\frac{3}{2}}}{d^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - 5 b c\right )}{d^{3} \left (a d - b c\right )} + \frac{\left (a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{\sqrt{b} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.28229, size = 134, normalized size = 0.81 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 c (7 b c-6 a d)}{(c+d x) (b c-a d)}-\frac{2 c^2}{(c+d x)^2}+3\right )}{3 d^3}+\frac{(a d-5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 \sqrt{b} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(3 - (2*c^2)/(c + d*x)^2 + (2*c*(7*b*c - 6*a*d))/((
b*c - a*d)*(c + d*x))))/(3*d^3) + ((-5*b*c + a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sq
rt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*Sqrt[b]*d^(7/2))

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Maple [B]  time = 0.03, size = 659, normalized size = 4. \[{\frac{1}{ \left ( 6\,ad-6\,bc \right ){d}^{3}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{d}^{4}-18\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}abc{d}^{3}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}{c}^{2}{d}^{2}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{a}^{2}c{d}^{3}-36\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xab{c}^{2}{d}^{2}+30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{3}d+6\,{x}^{2}a{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-6\,{x}^{2}bc{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{c}^{2}{d}^{2}-18\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ab{c}^{3}d+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{2}{c}^{4}+36\,xac{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-40\,xb{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+26\,a{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-30\,b{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(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^2*a^2*d^4-18*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^2*a*b*c*d^3+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^2*b^2*c^2*d^2+6*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*c*d^3-36*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^2*d^2+30*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^2*c^3*d+6*x
^2*a*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*x^2*b*c*d^2*((b*x+a)*(d*x+c))^(1/
2)*(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^2*c^2*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*b*c^3*d+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))*b^2*c^4+36*x*a*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)-40*x*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+26*a*c^2*d*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)-30*b*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))*(b*x+a)^
(1/2)/(a*d-b*c)/(b*d)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/d^3/(d*x+c)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.481259, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (15 \, b c^{3} - 13 \, a c^{2} d + 3 \,{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (10 \, b c^{2} d - 9 \, a c d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (5 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{12 \,{\left (b c^{3} d^{3} - a c^{2} d^{4} +{\left (b c d^{5} - a d^{6}\right )} x^{2} + 2 \,{\left (b c^{2} d^{4} - a c d^{5}\right )} x\right )} \sqrt{b d}}, \frac{2 \,{\left (15 \, b c^{3} - 13 \, a c^{2} d + 3 \,{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (10 \, b c^{2} d - 9 \, a c d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (5 \, b^{2} c^{4} - 6 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (5 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (5 \, b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{6 \,{\left (b c^{3} d^{3} - a c^{2} d^{4} +{\left (b c d^{5} - a d^{6}\right )} x^{2} + 2 \,{\left (b c^{2} d^{4} - a c d^{5}\right )} x\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(4*(15*b*c^3 - 13*a*c^2*d + 3*(b*c*d^2 - a*d^3)*x^2 + 2*(10*b*c^2*d - 9*a*
c*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(5*b^2*c^4 - 6*a*b*c^3*d + a
^2*c^2*d^2 + (5*b^2*c^2*d^2 - 6*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(5*b^2*c^3*d - 6*a*
b*c^2*d^2 + a^2*c*d^3)*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*c^3*d^3 - a*c^2*d^4 + (b*c*d^5 - a*d^6)*x^2 + 2*(b*c^2
*d^4 - a*c*d^5)*x)*sqrt(b*d)), 1/6*(2*(15*b*c^3 - 13*a*c^2*d + 3*(b*c*d^2 - a*d^
3)*x^2 + 2*(10*b*c^2*d - 9*a*c*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) -
3*(5*b^2*c^4 - 6*a*b*c^3*d + a^2*c^2*d^2 + (5*b^2*c^2*d^2 - 6*a*b*c*d^3 + a^2*d^
4)*x^2 + 2*(5*b^2*c^3*d - 6*a*b*c^2*d^2 + a^2*c*d^3)*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*c^3*d^3 - a*c^2*d^4
+ (b*c*d^5 - a*d^6)*x^2 + 2*(b*c^2*d^4 - a*c*d^5)*x)*sqrt(-b*d))]

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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(x**2*(b*x+a)**(1/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25389, size = 382, normalized size = 2.32 \[ \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{6} c d^{4} - a b^{5} d^{5}\right )}{\left (b x + a\right )}}{b^{4} c d^{5}{\left | b \right |} - a b^{3} d^{6}{\left | b \right |}} + \frac{2 \,{\left (10 \, b^{7} c^{2} d^{3} - 12 \, a b^{6} c d^{4} + 3 \, a^{2} b^{5} d^{5}\right )}}{b^{4} c d^{5}{\left | b \right |} - a b^{3} d^{6}{\left | b \right |}}\right )} + \frac{3 \,{\left (5 \, b^{8} c^{3} d^{2} - 11 \, a b^{7} c^{2} d^{3} + 7 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{4} c d^{5}{\left | b \right |} - a b^{3} d^{6}{\left | b \right |}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{{\left (5 \, b^{2} c - a b d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x + a)*(3*(b^6*c*d^4 - a*b^5*d^5)*(b*x + a)/(b^4*c*d^5*abs(b) - a*b^3*d^
6*abs(b)) + 2*(10*b^7*c^2*d^3 - 12*a*b^6*c*d^4 + 3*a^2*b^5*d^5)/(b^4*c*d^5*abs(b
) - a*b^3*d^6*abs(b))) + 3*(5*b^8*c^3*d^2 - 11*a*b^7*c^2*d^3 + 7*a^2*b^6*c*d^4 -
a^3*b^5*d^5)/(b^4*c*d^5*abs(b) - a*b^3*d^6*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x
+ a)*b*d - a*b*d)^(3/2) + (5*b^2*c - a*b*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + s
qrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d^3*abs(b))