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

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

[Out]

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

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Rubi [A]  time = 0.66095, 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 \[ \frac{(b c-a d)^3 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{11/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{128 b^5 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d) \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{192 b^4 d^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{240 b^3 d^2}-\frac{3 \sqrt{a+b x} (c+d x)^{7/2} (3 a d+b c)}{40 b^2 d^2}+\frac{x \sqrt{a+b x} (c+d x)^{7/2}}{5 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 55.8058, size = 301, normalized size = 0.96 \[ \frac{x \sqrt{a + b x} \left (c + d x\right )^{\frac{7}{2}}}{5 b d} - \frac{3 \sqrt{a + b x} \left (c + d x\right )^{\frac{7}{2}} \left (3 a d + b c\right )}{40 b^{2} d^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{240 b^{3} d^{2}} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{192 b^{4} d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{128 b^{5} d^{2}} - \frac{\left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{128 b^{\frac{11}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.26337, size = 257, normalized size = 0.82 \[ \frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{11/2} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (945 a^4 d^4-210 a^3 b d^3 (11 c+3 d x)+2 a^2 b^2 d^2 \left (782 c^2+749 c d x+252 d^2 x^2\right )-2 a b^3 d \left (45 c^3+481 c^2 d x+592 c d^2 x^2+216 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^5 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(945*a^4*d^4 - 210*a^3*b*d^3*(11*c + 3*d*x) + 2*a^2
*b^2*d^2*(782*c^2 + 749*c*d*x + 252*d^2*x^2) - 2*a*b^3*d*(45*c^3 + 481*c^2*d*x +
592*c*d^2*x^2 + 216*d^3*x^3) + b^4*(-45*c^4 + 30*c^3*d*x + 744*c^2*d^2*x^2 + 10
08*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^5*d^2) + ((b*c - a*d)^3*(3*b^2*c^2 + 14*a*
b*c*d + 63*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^(11/2)*d^(5/2))

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306508, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{4} - 45 \, b^{4} c^{4} - 90 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 2310 \, a^{3} b c d^{3} + 945 \, a^{4} d^{4} + 144 \,{\left (7 \, b^{4} c d^{3} - 3 \, a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (93 \, b^{4} c^{2} d^{2} - 148 \, a b^{3} c d^{3} + 63 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (15 \, b^{4} c^{3} d - 481 \, a b^{3} c^{2} d^{2} + 749 \, a^{2} b^{2} c d^{3} - 315 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{7680 \, \sqrt{b d} b^{5} d^{2}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{4} - 45 \, b^{4} c^{4} - 90 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 2310 \, a^{3} b c d^{3} + 945 \, a^{4} d^{4} + 144 \,{\left (7 \, b^{4} c d^{3} - 3 \, a b^{3} d^{4}\right )} x^{3} + 8 \,{\left (93 \, b^{4} c^{2} d^{2} - 148 \, a b^{3} c d^{3} + 63 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (15 \, b^{4} c^{3} d - 481 \, a b^{3} c^{2} d^{2} + 749 \, a^{2} b^{2} c d^{3} - 315 \, a^{3} b d^{4}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{3840 \, \sqrt{-b d} b^{5} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*b^4*d^4*x^4 - 45*b^4*c^4 - 90*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2
- 2310*a^3*b*c*d^3 + 945*a^4*d^4 + 144*(7*b^4*c*d^3 - 3*a*b^3*d^4)*x^3 + 8*(93*
b^4*c^2*d^2 - 148*a*b^3*c*d^3 + 63*a^2*b^2*d^4)*x^2 + 2*(15*b^4*c^3*d - 481*a*b^
3*c^2*d^2 + 749*a^2*b^2*c*d^3 - 315*a^3*b*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d
*x + c) - 15*(3*b^5*c^5 + 5*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^2*d
^3 + 175*a^4*b*c*d^4 - 63*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^5*d^2), 1/3840*(2*(384*b^4*d^4*x^4
- 45*b^4*c^4 - 90*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 2310*a^3*b*c*d^3 + 945*a^
4*d^4 + 144*(7*b^4*c*d^3 - 3*a*b^3*d^4)*x^3 + 8*(93*b^4*c^2*d^2 - 148*a*b^3*c*d^
3 + 63*a^2*b^2*d^4)*x^2 + 2*(15*b^4*c^3*d - 481*a*b^3*c^2*d^2 + 749*a^2*b^2*c*d^
3 - 315*a^3*b*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(3*b^5*c^5 + 5
*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^2*d^3 + 175*a^4*b*c*d^4 - 63*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^5*d^2)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.312158, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done