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

Optimal. Leaf size=217 \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{9/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (a d+7 b c)}{64 b d^4}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d) (a d+7 b c)}{96 b d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+7 b c)}{24 b d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b d} \]

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d^4) + (5*(b*
c - a*d)*(7*b*c + a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b*d^3) - ((7*b*c + a*d
)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*b*d^2) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4
*b*d) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(64*b^(3/2)*d^(9/2))

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Rubi [A]  time = 0.330674, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{9/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (a d+7 b c)}{64 b d^4}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d) (a d+7 b c)}{96 b d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+7 b c)}{24 b d^2}+\frac{(a+b x)^{7/2} \sqrt{c+d x}}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b*d^4) + (5*(b*
c - a*d)*(7*b*c + a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(96*b*d^3) - ((7*b*c + a*d
)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(24*b*d^2) + ((a + b*x)^(7/2)*Sqrt[c + d*x])/(4
*b*d) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(64*b^(3/2)*d^(9/2))

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Rubi in Sympy [A]  time = 33.7397, size = 196, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x}}{4 b d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d + 7 b c\right )}{24 b d^{2}} - \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (a d + 7 b c\right )}{96 b d^{3}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (a d + 7 b c\right )}{64 b d^{4}} - \frac{5 \left (a d - b c\right )^{3} \left (a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(7/2)*sqrt(c + d*x)/(4*b*d) - (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d + 7
*b*c)/(24*b*d**2) - 5*(a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)*(a*d + 7*b*c)/(
96*b*d**3) - 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2*(a*d + 7*b*c)/(64*b*d*
*4) - 5*(a*d - b*c)**3*(a*d + 7*b*c)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a
+ b*x)))/(64*b**(3/2)*d**(9/2))

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Mathematica [A]  time = 0.193818, size = 188, normalized size = 0.87 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (15 a^3 d^3+a^2 b d^2 (118 d x-191 c)+a b^2 d \left (265 c^2-172 c d x+136 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-56 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d^4}+\frac{5 (a d+7 b c) (b c-a d)^3 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{3/2} d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^3*d^3 + a^2*b*d^2*(-191*c + 118*d*x) + a*b^2*
d*(265*c^2 - 172*c*d*x + 136*d^2*x^2) + b^3*(-105*c^3 + 70*c^2*d*x - 56*c*d^2*x^
2 + 48*d^3*x^3)))/(192*b*d^4) + (5*(b*c - a*d)^3*(7*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]])/(128*b^(3/2)*d^(9/2))

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Maple [B]  time = 0.029, size = 574, normalized size = 2.7 \[ -{\frac{1}{384\,{d}^{4}b}\sqrt{bx+a}\sqrt{dx+c} \left ( -96\,{x}^{3}{b}^{3}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}-272\,{x}^{2}a{b}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+112\,{x}^{2}{b}^{3}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}+60\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}bc{d}^{3}-270\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{b}^{2}{c}^{2}{d}^{2}+300\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) a{b}^{3}d-105\,{c}^{4}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{4}-236\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{d}^{3}+344\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }xa{b}^{2}c{d}^{2}-140\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{b}^{3}{c}^{2}d-30\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{d}^{3}+382\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{2}bc{d}^{2}-530\,\sqrt{bd}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }a{b}^{2}{c}^{2}d+210\,{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*x^3*b^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)-272*x^2*a*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+112*x^2*b^3*c*d^2*
((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^4*d^4+60*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*b*c*d^3-270*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^2*c^2*d^2+300*c^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^3*
d-105*c^4*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^4-236*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b*d^3+344*(b*d)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)*x*a*b^2*c*d^2-140*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*b^
3*c^2*d-30*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^3+382*(b*d)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)*a^2*b*c*d^2-530*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*d+2
10*c^3*((b*x+a)*(d*x+c))^(1/2)*b^3*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/d^4/b/(b
*d)^(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((b*x + a)^(5/2)*x/sqrt(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276865, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 265 \, a b^{2} c^{2} d - 191 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 17 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 59 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{768 \, \sqrt{b d} b d^{4}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 265 \, a b^{2} c^{2} d - 191 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 17 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (35 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 59 \, a^{2} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{384 \, \sqrt{-b d} b d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*b^3*d^3*x^3 - 105*b^3*c^3 + 265*a*b^2*c^2*d - 191*a^2*b*c*d^2 + 15
*a^3*d^3 - 8*(7*b^3*c*d^2 - 17*a*b^2*d^3)*x^2 + 2*(35*b^3*c^2*d - 86*a*b^2*c*d^2
+ 59*a^2*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(7*b^4*c^4 - 20*a
*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*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)))/(sqrt(b*d)*b*d^4), 1/384
*(2*(48*b^3*d^3*x^3 - 105*b^3*c^3 + 265*a*b^2*c^2*d - 191*a^2*b*c*d^2 + 15*a^3*d
^3 - 8*(7*b^3*c*d^2 - 17*a*b^2*d^3)*x^2 + 2*(35*b^3*c^2*d - 86*a*b^2*c*d^2 + 59*
a^2*b*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(7*b^4*c^4 - 20*a*b^3*
c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*arctan(1/2*(2*b*d*x + b*c
+ a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b*d^4)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250527, size = 392, normalized size = 1.81 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2} d} - \frac{7 \, b^{3} c d^{5} + a b^{2} d^{6}}{b^{4} d^{7}}\right )} + \frac{5 \,{\left (7 \, b^{4} c^{2} d^{4} - 6 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} - \frac{15 \,{\left (7 \, b^{5} c^{3} d^{3} - 13 \, a b^{4} c^{2} d^{4} + 5 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} \sqrt{b x + a} - \frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b d^{4}}\right )} b}{192 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a
)/(b^2*d) - (7*b^3*c*d^5 + a*b^2*d^6)/(b^4*d^7)) + 5*(7*b^4*c^2*d^4 - 6*a*b^3*c*
d^5 - a^2*b^2*d^6)/(b^4*d^7)) - 15*(7*b^5*c^3*d^3 - 13*a*b^4*c^2*d^4 + 5*a^2*b^3
*c*d^5 + a^3*b^2*d^6)/(b^4*d^7))*sqrt(b*x + a) - 15*(7*b^4*c^4 - 20*a*b^3*c^3*d
+ 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*ln(abs(-sqrt(b*d)*sqrt(b*x + a)
+ sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^4))*b/abs(b)