3.579 \(\int \frac{x^3 \sqrt{a+b x}}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-2 b d x (35 b c-31 a d)-100 a b c d+105 b^2 c^2\right )}{12 b d^4 (b c-a d)}+\frac{\left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{9/2}}-\frac{2 x^2 \sqrt{a+b x} (7 b c-6 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 x^3 \sqrt{a+b x}}{3 d (c+d x)^{3/2}} \]

[Out]

(-2*x^3*Sqrt[a + b*x])/(3*d*(c + d*x)^(3/2)) - (2*(7*b*c - 6*a*d)*x^2*Sqrt[a + b
*x])/(3*d^2*(b*c - a*d)*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^2*c
^2 - 100*a*b*c*d + 3*a^2*d^2 - 2*b*d*(35*b*c - 31*a*d)*x))/(12*b*d^4*(b*c - a*d)
) + ((35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b
]*Sqrt[c + d*x])])/(4*b^(3/2)*d^(9/2))

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Rubi [A]  time = 0.525027, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-2 b d x (35 b c-31 a d)-100 a b c d+105 b^2 c^2\right )}{12 b d^4 (b c-a d)}+\frac{\left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{9/2}}-\frac{2 x^2 \sqrt{a+b x} (7 b c-6 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 x^3 \sqrt{a+b x}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*Sqrt[a + b*x])/(c + d*x)^(5/2),x]

[Out]

(-2*x^3*Sqrt[a + b*x])/(3*d*(c + d*x)^(3/2)) - (2*(7*b*c - 6*a*d)*x^2*Sqrt[a + b
*x])/(3*d^2*(b*c - a*d)*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^2*c
^2 - 100*a*b*c*d + 3*a^2*d^2 - 2*b*d*(35*b*c - 31*a*d)*x))/(12*b*d^4*(b*c - a*d)
) + ((35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b
]*Sqrt[c + d*x])])/(4*b^(3/2)*d^(9/2))

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Rubi in Sympy [A]  time = 42.13, size = 212, normalized size = 0.97 \[ - \frac{2 x^{3} \sqrt{a + b x}}{3 d \left (c + d x\right )^{\frac{3}{2}}} - \frac{2 x^{2} \sqrt{a + b x} \left (6 a d - 7 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )} + \frac{2 \sqrt{a + b x} \sqrt{c + d x} \left (\frac{3 a^{2} d^{2}}{8} - \frac{25 a b c d}{2} + \frac{105 b^{2} c^{2}}{8} + \frac{b d x \left (31 a d - 35 b c\right )}{4}\right )}{3 b d^{4} \left (a d - b c\right )} - \frac{\left (a^{2} d^{2} + 10 a b c d - 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{3}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)**(1/2)/(d*x+c)**(5/2),x)

[Out]

-2*x**3*sqrt(a + b*x)/(3*d*(c + d*x)**(3/2)) - 2*x**2*sqrt(a + b*x)*(6*a*d - 7*b
*c)/(3*d**2*sqrt(c + d*x)*(a*d - b*c)) + 2*sqrt(a + b*x)*sqrt(c + d*x)*(3*a**2*d
**2/8 - 25*a*b*c*d/2 + 105*b**2*c**2/8 + b*d*x*(31*a*d - 35*b*c)/4)/(3*b*d**4*(a
*d - b*c)) - (a**2*d**2 + 10*a*b*c*d - 35*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)
/(sqrt(b)*sqrt(c + d*x)))/(4*b**(3/2)*d**(9/2))

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Mathematica [A]  time = 0.356428, size = 164, normalized size = 0.75 \[ \frac{\left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{3/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{8 c^2 (10 b c-9 a d)}{(c+d x) (a d-b c)}+\frac{3 a d}{b}+\frac{8 c^3}{(c+d x)^2}-33 c+6 d x\right )}{12 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*Sqrt[a + b*x])/(c + d*x)^(5/2),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-33*c + (3*a*d)/b + 6*d*x + (8*c^3)/(c + d*x)^2 +
(8*c^2*(10*b*c - 9*a*d))/((-(b*c) + a*d)*(c + d*x))))/(12*d^4) + ((35*b^2*c^2 -
10*a*b*c*d - a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*
Sqrt[c + d*x]])/(8*b^(3/2)*d^(9/2))

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Maple [B]  time = 0.043, size = 986, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(1/2)/(d*x+c)^(5/2),x)

[Out]

-1/24*(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))*x^2*a^3*d^5+27*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))*x^2*a^2*b*c*d^4-135*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))*x^2*a*b^2*c^2*d^3+105*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))*x^2*b^3*c^3*d^2-12*x^3*a*b*d^4
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+12*x^3*b^2*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+6*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))*x*a^3*c*d^4+54*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))*x*a^2*b*c^2*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))*x*a*b^2*c^3*d^2+210*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))*x*b^3*c^4*d-6*x^2*a^2*d^4*((b
*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+48*x^2*a*b*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)-42*x^2*b^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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^3*c^2*d^3+27*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*c^3*
d^2-135*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^2*c^4*d+105*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^3*c^5-12*x*a^2*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+276*x
*a*b*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-280*x*b^2*c^3*d*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)-6*a^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+200*a*b*c
^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-210*b^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2))*(b*x+a)^(1/2)/(a*d-b*c)/(b*d)^(1/2)/b/((b*x+a)*(d*x+c))^(1/2)/d^4/(d*x
+c)^(3/2)

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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(sqrt(b*x + a)*x^3/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.676638, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*x^3/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(4*(105*b^2*c^4 - 100*a*b*c^3*d + 3*a^2*c^2*d^2 - 6*(b^2*c*d^3 - a*b*d^4)
*x^3 + 3*(7*b^2*c^2*d^2 - 8*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(70*b^2*c^3*d - 69*a*b*
c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(35*b^3*c^5
- 45*a*b^2*c^4*d + 9*a^2*b*c^3*d^2 + a^3*c^2*d^3 + (35*b^3*c^3*d^2 - 45*a*b^2*c^
2*d^3 + 9*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(35*b^3*c^4*d - 45*a*b^2*c^3*d^2 + 9*a^
2*b*c^2*d^3 + a^3*c*d^4)*x)*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)))/((b^2*c^3*d^4 - a*b*c^2*d^5 + (b^2*c*d^6 - a*b*d^7)*x^2
+ 2*(b^2*c^2*d^5 - a*b*c*d^6)*x)*sqrt(b*d)), -1/24*(2*(105*b^2*c^4 - 100*a*b*c^
3*d + 3*a^2*c^2*d^2 - 6*(b^2*c*d^3 - a*b*d^4)*x^3 + 3*(7*b^2*c^2*d^2 - 8*a*b*c*d
^3 + a^2*d^4)*x^2 + 2*(70*b^2*c^3*d - 69*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(-b*d
)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(35*b^3*c^5 - 45*a*b^2*c^4*d + 9*a^2*b*c^3*d^2
+ a^3*c^2*d^3 + (35*b^3*c^3*d^2 - 45*a*b^2*c^2*d^3 + 9*a^2*b*c*d^4 + a^3*d^5)*x
^2 + 2*(35*b^3*c^4*d - 45*a*b^2*c^3*d^2 + 9*a^2*b*c^2*d^3 + a^3*c*d^4)*x)*arctan
(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/((b^2*
c^3*d^4 - a*b*c^2*d^5 + (b^2*c*d^6 - a*b*d^7)*x^2 + 2*(b^2*c^2*d^5 - a*b*c*d^6)*
x)*sqrt(-b*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274925, size = 549, normalized size = 2.52 \[ \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{5} c d^{6}{\left | b \right |} - a b^{4} d^{7}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{6} c d^{7} - a b^{5} d^{8}} - \frac{7 \, b^{6} c^{2} d^{5}{\left | b \right |} - 2 \, a b^{5} c d^{6}{\left | b \right |} - 5 \, a^{2} b^{4} d^{7}{\left | b \right |}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )} - \frac{4 \,{\left (35 \, b^{7} c^{3} d^{4}{\left | b \right |} - 45 \, a b^{6} c^{2} d^{5}{\left | b \right |} + 9 \, a^{2} b^{5} c d^{6}{\left | b \right |} + 3 \, a^{3} b^{4} d^{7}{\left | b \right |}\right )}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )}{\left (b x + a\right )} - \frac{3 \,{\left (35 \, b^{8} c^{4} d^{3}{\left | b \right |} - 80 \, a b^{7} c^{3} d^{4}{\left | b \right |} + 54 \, a^{2} b^{6} c^{2} d^{5}{\left | b \right |} - 8 \, a^{3} b^{5} c d^{6}{\left | b \right |} - a^{4} b^{4} d^{7}{\left | b \right |}\right )}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{{\left (35 \, b^{2} c^{2}{\left | b \right |} - 10 \, a b c d{\left | b \right |} - a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} b^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*x^3/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

1/12*((3*(b*x + a)*(2*(b^5*c*d^6*abs(b) - a*b^4*d^7*abs(b))*(b*x + a)/(b^6*c*d^7
- a*b^5*d^8) - (7*b^6*c^2*d^5*abs(b) - 2*a*b^5*c*d^6*abs(b) - 5*a^2*b^4*d^7*abs
(b))/(b^6*c*d^7 - a*b^5*d^8)) - 4*(35*b^7*c^3*d^4*abs(b) - 45*a*b^6*c^2*d^5*abs(
b) + 9*a^2*b^5*c*d^6*abs(b) + 3*a^3*b^4*d^7*abs(b))/(b^6*c*d^7 - a*b^5*d^8))*(b*
x + a) - 3*(35*b^8*c^4*d^3*abs(b) - 80*a*b^7*c^3*d^4*abs(b) + 54*a^2*b^6*c^2*d^5
*abs(b) - 8*a^3*b^5*c*d^6*abs(b) - a^4*b^4*d^7*abs(b))/(b^6*c*d^7 - a*b^5*d^8))*
sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 1/4*(35*b^2*c^2*abs(b) - 1
0*a*b*c*d*abs(b) - a^2*d^2*abs(b))*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^2*d^4)