3.843 \(\int \frac{(d+e x)^{3/2} \left (15 d^2+20 d e x+8 e^2 x^2\right )}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=240 \[ \frac{(b d-a e)^2 \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{9/2} \sqrt{e}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{8 b^4}+\frac{\sqrt{a+b x} (d+e x)^{3/2} \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{12 b^3}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\sqrt{a+b x} (d+e x)^{5/2} (17 b d-13 a e)}{3 b^2} \]

[Out]

((b*d - a*e)*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x]*Sqrt[d + e*x])
/(8*b^4) + ((73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x]*(d + e*x)^(3/2)
)/(12*b^3) + ((17*b*d - 13*a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^2) + (2*e*(a
+ b*x)^(3/2)*(d + e*x)^(5/2))/b^2 + ((b*d - a*e)^2*(73*b^2*d^2 - 90*a*b*d*e + 3
5*a^2*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(9/2)*
Sqrt[e])

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Rubi [A]  time = 0.485785, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132 \[ \frac{(b d-a e)^2 \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{9/2} \sqrt{e}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{8 b^4}+\frac{\sqrt{a+b x} (d+e x)^{3/2} \left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right )}{12 b^3}+\frac{2 e (a+b x)^{3/2} (d+e x)^{5/2}}{b^2}+\frac{\sqrt{a+b x} (d+e x)^{5/2} (17 b d-13 a e)}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(3/2)*(15*d^2 + 20*d*e*x + 8*e^2*x^2))/Sqrt[a + b*x],x]

[Out]

((b*d - a*e)*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x]*Sqrt[d + e*x])
/(8*b^4) + ((73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*Sqrt[a + b*x]*(d + e*x)^(3/2)
)/(12*b^3) + ((17*b*d - 13*a*e)*Sqrt[a + b*x]*(d + e*x)^(5/2))/(3*b^2) + (2*e*(a
+ b*x)^(3/2)*(d + e*x)^(5/2))/b^2 + ((b*d - a*e)^2*(73*b^2*d^2 - 90*a*b*d*e + 3
5*a^2*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(9/2)*
Sqrt[e])

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Rubi in Sympy [A]  time = 75.985, size = 396, normalized size = 1.65 \[ \frac{20 d \sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}}}{3 b} + \frac{2 e x \sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}}}{b} - \frac{5 d \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (10 a e - 7 b d\right )}{6 b^{2}} - \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}} \left (7 a e + 3 b d\right )}{3 b^{2}} + \frac{5 d \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (10 a e - 7 b d\right )}{4 b^{3}} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (35 a^{2} e^{2} + 10 a b d e + 3 b^{2} d^{2}\right )}{12 b^{3}} - \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (35 a^{2} e^{2} + 10 a b d e + 3 b^{2} d^{2}\right )}{8 b^{4}} - \frac{5 d \left (a e - b d\right )^{2} \left (10 a e - 7 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{4 b^{\frac{7}{2}} \sqrt{e}} + \frac{\left (a e - b d\right )^{2} \left (35 a^{2} e^{2} + 10 a b d e + 3 b^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{8 b^{\frac{9}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(8*e**2*x**2+20*d*e*x+15*d**2)/(b*x+a)**(1/2),x)

[Out]

20*d*sqrt(a + b*x)*(d + e*x)**(5/2)/(3*b) + 2*e*x*sqrt(a + b*x)*(d + e*x)**(5/2)
/b - 5*d*sqrt(a + b*x)*(d + e*x)**(3/2)*(10*a*e - 7*b*d)/(6*b**2) - sqrt(a + b*x
)*(d + e*x)**(5/2)*(7*a*e + 3*b*d)/(3*b**2) + 5*d*sqrt(a + b*x)*sqrt(d + e*x)*(a
*e - b*d)*(10*a*e - 7*b*d)/(4*b**3) + sqrt(a + b*x)*(d + e*x)**(3/2)*(35*a**2*e*
*2 + 10*a*b*d*e + 3*b**2*d**2)/(12*b**3) - sqrt(a + b*x)*sqrt(d + e*x)*(a*e - b*
d)*(35*a**2*e**2 + 10*a*b*d*e + 3*b**2*d**2)/(8*b**4) - 5*d*(a*e - b*d)**2*(10*a
*e - 7*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(4*b**(7/2)*sqr
t(e)) + (a*e - b*d)**2*(35*a**2*e**2 + 10*a*b*d*e + 3*b**2*d**2)*atanh(sqrt(e)*s
qrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(8*b**(9/2)*sqrt(e))

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Mathematica [A]  time = 0.223334, size = 202, normalized size = 0.84 \[ \frac{\left (35 a^2 e^2-90 a b d e+73 b^2 d^2\right ) (b d-a e)^2 \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 b^{9/2} \sqrt{e}}+\frac{\sqrt{a+b x} \sqrt{d+e x} \left (-105 a^3 e^3+5 a^2 b e^2 (89 d+14 e x)-a b^2 e \left (725 d^2+292 d e x+56 e^2 x^2\right )+b^3 \left (501 d^3+466 d^2 e x+232 d e^2 x^2+48 e^3 x^3\right )\right )}{24 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(3/2)*(15*d^2 + 20*d*e*x + 8*e^2*x^2))/Sqrt[a + b*x],x]

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^3*e^3 + 5*a^2*b*e^2*(89*d + 14*e*x) - a*b^2
*e*(725*d^2 + 292*d*e*x + 56*e^2*x^2) + b^3*(501*d^3 + 466*d^2*e*x + 232*d*e^2*x
^2 + 48*e^3*x^3)))/(24*b^4) + ((b*d - a*e)^2*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e
^2)*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*Sqrt[d + e*x]])/(1
6*b^(9/2)*Sqrt[e])

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Maple [B]  time = 0.049, size = 571, normalized size = 2.4 \[{\frac{1}{48\,{b}^{4}}\sqrt{ex+d}\sqrt{bx+a} \left ( 96\,{x}^{3}{b}^{3}{e}^{3}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-112\,{x}^{2}a{b}^{2}{e}^{3}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+464\,{x}^{2}{b}^{3}d{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+105\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{4}{e}^{4}-480\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{3}{e}^{3}db+864\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}{e}^{2}{d}^{2}{b}^{2}-708\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) ae{d}^{3}{b}^{3}+219\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){d}^{4}{b}^{4}+140\,{e}^{3}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }x{a}^{2}b\sqrt{be}-584\,{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xad{b}^{2}\sqrt{be}+932\,e\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }x{d}^{2}{b}^{3}\sqrt{be}-210\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{a}^{3}{e}^{3}\sqrt{be}+890\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{a}^{2}d{e}^{2}b\sqrt{be}-1450\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }a{d}^{2}e{b}^{2}\sqrt{be}+1002\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{d}^{3}{b}^{3}\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(8*e^2*x^2+20*d*e*x+15*d^2)/(b*x+a)^(1/2),x)

[Out]

1/48*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(96*x^3*b^3*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)-112*x^2*a*b^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+464*x^2*b^3*d*e^2*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+105*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*e^4-480*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*e^3*d*b+864*ln(1/2*(2*b*e*x+2*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*e^2*d^2*b^2-708*ln(1/2*
(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*e*d^3*b^3
+219*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))
*d^4*b^4+140*e^3*((b*x+a)*(e*x+d))^(1/2)*x*a^2*b*(b*e)^(1/2)-584*e^2*((b*x+a)*(e
*x+d))^(1/2)*x*a*d*b^2*(b*e)^(1/2)+932*e*((b*x+a)*(e*x+d))^(1/2)*x*d^2*b^3*(b*e)
^(1/2)-210*((b*x+a)*(e*x+d))^(1/2)*a^3*e^3*(b*e)^(1/2)+890*((b*x+a)*(e*x+d))^(1/
2)*a^2*d*e^2*b*(b*e)^(1/2)-1450*((b*x+a)*(e*x+d))^(1/2)*a*d^2*e*b^2*(b*e)^(1/2)+
1002*((b*x+a)*(e*x+d))^(1/2)*d^3*b^3*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/b^4/(b
*e)^(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((8*e^2*x^2 + 20*d*e*x + 15*d^2)*(e*x + d)^(3/2)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.343964, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} e^{3} x^{3} + 501 \, b^{3} d^{3} - 725 \, a b^{2} d^{2} e + 445 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \,{\left (29 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (233 \, b^{3} d^{2} e - 146 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} + 3 \,{\left (73 \, b^{4} d^{4} - 236 \, a b^{3} d^{3} e + 288 \, a^{2} b^{2} d^{2} e^{2} - 160 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{96 \, \sqrt{b e} b^{4}}, \frac{2 \,{\left (48 \, b^{3} e^{3} x^{3} + 501 \, b^{3} d^{3} - 725 \, a b^{2} d^{2} e + 445 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \,{\left (29 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (233 \, b^{3} d^{2} e - 146 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} + 3 \,{\left (73 \, b^{4} d^{4} - 236 \, a b^{3} d^{3} e + 288 \, a^{2} b^{2} d^{2} e^{2} - 160 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{48 \, \sqrt{-b e} b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)*(e*x + d)^(3/2)/sqrt(b*x + a),x, algorithm="fricas")

[Out]

[1/96*(4*(48*b^3*e^3*x^3 + 501*b^3*d^3 - 725*a*b^2*d^2*e + 445*a^2*b*d*e^2 - 105
*a^3*e^3 + 8*(29*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + 2*(233*b^3*d^2*e - 146*a*b^2*d*e
^2 + 35*a^2*b*e^3)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(73*b^4*d^4 - 23
6*a*b^3*d^3*e + 288*a^2*b^2*d^2*e^2 - 160*a^3*b*d*e^3 + 35*a^4*e^4)*log(4*(2*b^2
*e^2*x + b^2*d*e + a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2*e^2*x^2 + b^2*d
^2 + 6*a*b*d*e + a^2*e^2 + 8*(b^2*d*e + a*b*e^2)*x)*sqrt(b*e)))/(sqrt(b*e)*b^4),
1/48*(2*(48*b^3*e^3*x^3 + 501*b^3*d^3 - 725*a*b^2*d^2*e + 445*a^2*b*d*e^2 - 105
*a^3*e^3 + 8*(29*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + 2*(233*b^3*d^2*e - 146*a*b^2*d*e
^2 + 35*a^2*b*e^3)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(73*b^4*d^4 - 2
36*a*b^3*d^3*e + 288*a^2*b^2*d^2*e^2 - 160*a^3*b*d*e^3 + 35*a^4*e^4)*arctan(1/2*
(2*b*e*x + b*d + a*e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/(sqrt(-b*e)
*b^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((e*x+d)**(3/2)*(8*e**2*x**2+20*d*e*x+15*d**2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.380714, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)*(e*x + d)^(3/2)/sqrt(b*x + a),x, algorithm="giac")

[Out]

Done