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

Optimal. Leaf size=376 \[ -\frac{\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{11/2}}+\frac{(a+b x)^{7/2} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right )}{160 b^3 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^3}{512 b^3 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^2}{768 b^3 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)}{960 b^3 d^3}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+9 b c)}{60 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d} \]

[Out]

((b*c - a*d)^3*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(512*b^3*d^5) - ((b*c - a*d)^2*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*(a + b*x)
^(3/2)*Sqrt[c + d*x])/(768*b^3*d^4) + ((b*c - a*d)*(21*b^2*c^2 + 14*a*b*c*d + 5*
a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(960*b^3*d^3) + ((21*b^2*c^2 + 14*a*b*c*
d + 5*a^2*d^2)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(160*b^3*d^2) - ((9*b*c + 5*a*d)*(
a + b*x)^(7/2)*(c + d*x)^(3/2))/(60*b^2*d^2) + (x*(a + b*x)^(7/2)*(c + d*x)^(3/2
))/(6*b*d) - ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[
d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(11/2))

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Rubi [A]  time = 0.834345, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{11/2}}+\frac{(a+b x)^{7/2} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right )}{160 b^3 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^3}{512 b^3 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^2}{768 b^3 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)}{960 b^3 d^3}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+9 b c)}{60 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{3/2}}{6 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^3*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(512*b^3*d^5) - ((b*c - a*d)^2*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*(a + b*x)
^(3/2)*Sqrt[c + d*x])/(768*b^3*d^4) + ((b*c - a*d)*(21*b^2*c^2 + 14*a*b*c*d + 5*
a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(960*b^3*d^3) + ((21*b^2*c^2 + 14*a*b*c*
d + 5*a^2*d^2)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(160*b^3*d^2) - ((9*b*c + 5*a*d)*(
a + b*x)^(7/2)*(c + d*x)^(3/2))/(60*b^2*d^2) + (x*(a + b*x)^(7/2)*(c + d*x)^(3/2
))/(6*b*d) - ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[
d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(11/2))

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Rubi in Sympy [A]  time = 74.9762, size = 360, normalized size = 0.96 \[ \frac{x \left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}}}{6 b d} - \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + 9 b c\right )}{60 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (5 a^{2} d^{2} + 14 a b c d + 21 b^{2} c^{2}\right )}{160 b^{3} d^{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 14 a b c d + 21 b^{2} c^{2}\right )}{960 b^{3} d^{3}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (5 a^{2} d^{2} + 14 a b c d + 21 b^{2} c^{2}\right )}{768 b^{3} d^{4}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (5 a^{2} d^{2} + 14 a b c d + 21 b^{2} c^{2}\right )}{512 b^{3} d^{5}} - \frac{\left (a d - b c\right )^{4} \left (5 a^{2} d^{2} + 14 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{512 b^{\frac{7}{2}} d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x)**(7/2)*(c + d*x)**(3/2)/(6*b*d) - (a + b*x)**(7/2)*(c + d*x)**(3/2)*
(5*a*d + 9*b*c)/(60*b**2*d**2) + (a + b*x)**(7/2)*sqrt(c + d*x)*(5*a**2*d**2 + 1
4*a*b*c*d + 21*b**2*c**2)/(160*b**3*d**2) - (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d
- b*c)*(5*a**2*d**2 + 14*a*b*c*d + 21*b**2*c**2)/(960*b**3*d**3) - (a + b*x)**(3
/2)*sqrt(c + d*x)*(a*d - b*c)**2*(5*a**2*d**2 + 14*a*b*c*d + 21*b**2*c**2)/(768*
b**3*d**4) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3*(5*a**2*d**2 + 14*a*b*c*
d + 21*b**2*c**2)/(512*b**3*d**5) - (a*d - b*c)**4*(5*a**2*d**2 + 14*a*b*c*d + 2
1*b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x)))/(512*b**(7/2)*
d**(11/2))

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Mathematica [A]  time = 0.291157, size = 319, normalized size = 0.85 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (75 a^5 d^5-5 a^4 b d^4 (13 c+10 d x)+10 a^3 b^2 d^3 \left (-9 c^2+4 c d x+4 d^2 x^2\right )+2 a^2 b^3 d^2 \left (419 c^3-262 c^2 d x+204 c d^2 x^2+1080 d^3 x^3\right )+a b^4 d \left (-945 c^4+616 c^3 d x-488 c^2 d^2 x^2+416 c d^3 x^3+3200 d^4 x^4\right )+b^5 \left (315 c^5-210 c^4 d x+168 c^3 d^2 x^2-144 c^2 d^3 x^3+128 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^3 d^5}-\frac{(b c-a d)^4 \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{1024 b^{7/2} d^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(75*a^5*d^5 - 5*a^4*b*d^4*(13*c + 10*d*x) + 10*a^3*
b^2*d^3*(-9*c^2 + 4*c*d*x + 4*d^2*x^2) + 2*a^2*b^3*d^2*(419*c^3 - 262*c^2*d*x +
204*c*d^2*x^2 + 1080*d^3*x^3) + a*b^4*d*(-945*c^4 + 616*c^3*d*x - 488*c^2*d^2*x^
2 + 416*c*d^3*x^3 + 3200*d^4*x^4) + b^5*(315*c^5 - 210*c^4*d*x + 168*c^3*d^2*x^2
- 144*c^2*d^3*x^3 + 128*c*d^4*x^4 + 1280*d^5*x^5)))/(7680*b^3*d^5) - ((b*c - a*
d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*S
qrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(1024*b^(7/2)*d^(11/2))

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Maple [B]  time = 0.027, size = 1240, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-80*x^2*a^3*b^2*d^5*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*(b*d)^(1/2)-336*x^2*b^5*c^3*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^
(1/2)-6400*x^4*a*b^4*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-256*x^4*b^5
*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-4320*x^3*a^2*b^3*d^5*(b*d*x^2
+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+288*x^3*b^5*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*(b*d)^(1/2)-1676*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^3*d^2*(b*d)^(
1/2)+1890*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*d*(b*d)^(1/2)+100*d^5*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*b*(b*d)^(1/2)+420*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*x*b^5*d*(b*d)^(1/2)+130*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*b*(b*d)^
(1/2)+180*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*d^3*(b*d)^(1/2)+75*d^6*ln(
1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))
*a^6+315*c^6*b^6*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))-2560*x^5*b^5*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2
)-150*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*(b*d)^(1/2)-630*c^5*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*b^5*(b*d)^(1/2)-90*d^5*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*c*b-75*c^2*d^4*ln(1/2*(2*b*d*x+
2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2-300*
c^3*a^3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(
b*d)^(1/2))*b^3*d^3+1125*c^4*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(
b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*d^2-1050*c^5*a*ln(1/2*(2*b*d*x+2*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^5*d-80*d^4*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*c*b^2*(b*d)^(1/2)+1048*c^2*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*x*a^2*b^3*d^3*(b*d)^(1/2)-1232*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a
*b^4*d^2*(b*d)^(1/2)-816*x^2*a^2*b^3*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)
^(1/2)+976*x^2*a*b^4*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-832*x^3
*a*b^4*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)/d^5/b^3/(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)*sqrt(d*x + c)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289599, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(4*(1280*b^5*d^5*x^5 + 315*b^5*c^5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*
d^2 - 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 75*a^5*d^5 + 128*(b^5*c*d^4 + 25*a*b
^4*d^5)*x^4 - 16*(9*b^5*c^2*d^3 - 26*a*b^4*c*d^4 - 135*a^2*b^3*d^5)*x^3 + 8*(21*
b^5*c^3*d^2 - 61*a*b^4*c^2*d^3 + 51*a^2*b^3*c*d^4 + 5*a^3*b^2*d^5)*x^2 - 2*(105*
b^5*c^4*d - 308*a*b^4*c^3*d^2 + 262*a^2*b^3*c^2*d^3 - 20*a^3*b^2*c*d^4 + 25*a^4*
b*d^5)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(21*b^6*c^6 - 70*a*b^5*c^5*
d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5
+ 5*a^6*d^6)*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)*sq
rt(b*d)))/(sqrt(b*d)*b^3*d^5), 1/15360*(2*(1280*b^5*d^5*x^5 + 315*b^5*c^5 - 945*
a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 75*a^5
*d^5 + 128*(b^5*c*d^4 + 25*a*b^4*d^5)*x^4 - 16*(9*b^5*c^2*d^3 - 26*a*b^4*c*d^4 -
135*a^2*b^3*d^5)*x^3 + 8*(21*b^5*c^3*d^2 - 61*a*b^4*c^2*d^3 + 51*a^2*b^3*c*d^4
+ 5*a^3*b^2*d^5)*x^2 - 2*(105*b^5*c^4*d - 308*a*b^4*c^3*d^2 + 262*a^2*b^3*c^2*d^
3 - 20*a^3*b^2*c*d^4 + 25*a^4*b*d^5)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) -
15*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a
^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + 5*a^6*d^6)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqr
t(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b^3*d^5)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.313864, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done