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

Optimal. Leaf size=314 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{128 b^2 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{192 b^2 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{240 b^2 d^3}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{11/2}}-\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (a d+3 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d} \]

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(128*b^2*d^5) - ((b*c - a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(
3/2)*Sqrt[c + d*x])/(192*b^2*d^4) + ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a +
b*x)^(5/2)*Sqrt[c + d*x])/(240*b^2*d^3) - (3*(3*b*c + a*d)*(a + b*x)^(7/2)*Sqrt[
c + d*x])/(40*b^2*d^2) + (x*(a + b*x)^(7/2)*Sqrt[c + d*x])/(5*b*d) - ((b*c - a*d
)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[
b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(11/2))

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Rubi [A]  time = 0.669547, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{128 b^2 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{192 b^2 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{240 b^2 d^3}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{5/2} d^{11/2}}-\frac{3 (a+b x)^{7/2} \sqrt{c+d x} (a d+3 b c)}{40 b^2 d^2}+\frac{x (a+b x)^{7/2} \sqrt{c+d x}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(128*b^2*d^5) - ((b*c - a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*x)^(
3/2)*Sqrt[c + d*x])/(192*b^2*d^4) + ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a +
b*x)^(5/2)*Sqrt[c + d*x])/(240*b^2*d^3) - (3*(3*b*c + a*d)*(a + b*x)^(7/2)*Sqrt[
c + d*x])/(40*b^2*d^2) + (x*(a + b*x)^(7/2)*Sqrt[c + d*x])/(5*b*d) - ((b*c - a*d
)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[
b]*Sqrt[c + d*x])])/(128*b^(5/2)*d^(11/2))

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Rubi in Sympy [A]  time = 55.5684, size = 301, normalized size = 0.96 \[ \frac{x \left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x}}{5 b d} - \frac{3 \left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (a d + 3 b c\right )}{40 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{240 b^{2} d^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{192 b^{2} d^{4}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{128 b^{2} d^{5}} + \frac{\left (a d - b c\right )^{3} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{128 b^{\frac{5}{2}} d^{\frac{11}{2}}} \]

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)*sqrt(c + d*x)/(5*b*d) - 3*(a + b*x)**(7/2)*sqrt(c + d*x)*(a*d
+ 3*b*c)/(40*b**2*d**2) + (a + b*x)**(5/2)*sqrt(c + d*x)*(3*a**2*d**2 + 14*a*b*
c*d + 63*b**2*c**2)/(240*b**2*d**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)
*(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)/(192*b**2*d**4) + sqrt(a + b*x)*sqrt(
c + d*x)*(a*d - b*c)**2*(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)/(128*b**2*d**5
) + (a*d - b*c)**3*(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)*atanh(sqrt(b)*sqrt(
c + d*x)/(sqrt(d)*sqrt(a + b*x)))/(128*b**(5/2)*d**(11/2))

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Mathematica [A]  time = 0.261616, size = 256, normalized size = 0.82 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (d x-3 c)+2 a^2 b^2 d^2 \left (782 c^2-481 c d x+372 d^2 x^2\right )+2 a b^3 d \left (-1155 c^3+749 c^2 d x-592 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (945 c^4-630 c^3 d x+504 c^2 d^2 x^2-432 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^5}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{5/2} d^{11/2}} \]

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]*(-45*a^4*d^4 + 30*a^3*b*d^3*(-3*c + d*x) + 2*a^2*b^
2*d^2*(782*c^2 - 481*c*d*x + 372*d^2*x^2) + 2*a*b^3*d*(-1155*c^3 + 749*c^2*d*x -
592*c*d^2*x^2 + 504*d^3*x^3) + b^4*(945*c^4 - 630*c^3*d*x + 504*c^2*d^2*x^2 - 4
32*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^2*d^5) - ((b*c - a*d)^3*(63*b^2*c^2 + 14*a
*b*c*d + 3*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sq
rt[c + d*x]])/(256*b^(5/2)*d^(11/2))

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Maple [B]  time = 0.036, size = 788, normalized size = 2.5 \[ \text{result too large to display} \]

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]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*x^4*b^4*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)+2016*x^3*a*b^3*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-864*x^3*b^4*c*d^3
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1488*x^2*a^2*b^2*d^4*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)-2368*x^2*a*b^3*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1008*x^2*
b^4*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+45*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^5*d^5+75*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*b*c*d^4+450*c^2*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^3*b^
2-2250*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^2*b^3*d^2+2625*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))*a*b^4*d-945*c^5*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^5+60*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x
*a^3*b*d^4-1924*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b^2*c*d^3+2996*(b*d)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a*b^3*c^2*d^2-1260*(b*d)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)*x*b^4*c^3*d-90*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*d^4-180*(b*d)^(1/2)
*((b*x+a)*(d*x+c))^(1/2)*a^3*b*c*d^3+3128*c^2*((b*x+a)*(d*x+c))^(1/2)*a^2*d^2*b^
2*(b*d)^(1/2)-4620*c^3*((b*x+a)*(d*x+c))^(1/2)*a*b^3*d*(b*d)^(1/2)+1890*c^4*((b*
x+a)*(d*x+c))^(1/2)*b^4*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/d^5/b^2/(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^2/sqrt(d*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297848, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} + 945 \, b^{4} c^{4} - 2310 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 90 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4} - 144 \,{\left (3 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (63 \, b^{4} c^{2} d^{2} - 148 \, a b^{3} c d^{3} + 93 \, a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (315 \, b^{4} c^{3} d - 749 \, a b^{3} c^{2} d^{2} + 481 \, a^{2} b^{2} c d^{3} - 15 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b^{2} d^{5}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} + 945 \, b^{4} c^{4} - 2310 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 90 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4} - 144 \,{\left (3 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (63 \, b^{4} c^{2} d^{2} - 148 \, a b^{3} c d^{3} + 93 \, a^{2} b^{2} d^{4}\right )} x^{2} - 2 \,{\left (315 \, b^{4} c^{3} d - 749 \, a b^{3} c^{2} d^{2} + 481 \, a^{2} b^{2} c d^{3} - 15 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b^{2} d^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*b^4*d^4*x^4 + 945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*
d^2 - 90*a^3*b*c*d^3 - 45*a^4*d^4 - 144*(3*b^4*c*d^3 - 7*a*b^3*d^4)*x^3 + 8*(63*
b^4*c^2*d^2 - 148*a*b^3*c*d^3 + 93*a^2*b^2*d^4)*x^2 - 2*(315*b^4*c^3*d - 749*a*b
^3*c^2*d^2 + 481*a^2*b^2*c*d^3 - 15*a^3*b*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d
*x + c) - 15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^
2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*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^2*d^5), 1/3840*(2*(384*b^4*d^4*x^4 +
945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 90*a^3*b*c*d^3 - 45*a^4
*d^4 - 144*(3*b^4*c*d^3 - 7*a*b^3*d^4)*x^3 + 8*(63*b^4*c^2*d^2 - 148*a*b^3*c*d^3
+ 93*a^2*b^2*d^4)*x^2 - 2*(315*b^4*c^3*d - 749*a*b^3*c^2*d^2 + 481*a^2*b^2*c*d^
3 - 15*a^3*b*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(63*b^5*c^5 - 1
75*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^
5*d^5)*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^2*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 = 0.262643, size = 516, normalized size = 1.64 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (4 \,{\left (b x + a\right )}{\left (6 \,{\left (b x + a\right )}{\left (\frac{8 \,{\left (b x + a\right )}}{b^{3} d} - \frac{9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac{63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac{5 \,{\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} \sqrt{b x + a} + \frac{15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{5}}\right )} b}{1920 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x
+ a)/(b^3*d) - (9*b^7*c*d^7 + 11*a*b^6*d^8)/(b^9*d^9)) + (63*b^8*c^2*d^6 + 14*a
*b^7*c*d^7 + 3*a^2*b^6*d^8)/(b^9*d^9)) - 5*(63*b^9*c^3*d^5 - 49*a*b^8*c^2*d^6 -
11*a^2*b^7*c*d^7 - 3*a^3*b^6*d^8)/(b^9*d^9))*(b*x + a) + 15*(63*b^10*c^4*d^4 - 1
12*a*b^9*c^3*d^5 + 38*a^2*b^8*c^2*d^6 + 8*a^3*b^7*c*d^7 + 3*a^4*b^6*d^8)/(b^9*d^
9))*sqrt(b*x + a) + 15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*
a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*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)*b^2*d^5))*b/abs(b)