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

Optimal. Leaf size=306 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-\frac{63 a^2 d}{b}+14 a c+\frac{b c^2}{d}\right )}{24 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 \sqrt{a+b x} (b c-a d)}-\frac{5 (b c-a d)^2 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{64 b^5 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{96 b^4 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d} \]

[Out]

(-5*(b*c - a*d)*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(64*b^5*d) - (5*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/
2))/(96*b^4*d) - ((14*a*c + (b*c^2)/d - (63*a^2*d)/b)*Sqrt[a + b*x]*(c + d*x)^(5
/2))/(24*b^2*(b*c - a*d)) - (2*a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*Sqrt[a + b*
x]) + (Sqrt[a + b*x]*(c + d*x)^(7/2))/(4*b^2*d) - (5*(b*c - a*d)^2*(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])])
/(64*b^(11/2)*d^(3/2))

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Rubi [A]  time = 0.745691, antiderivative size = 306, 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 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-\frac{63 a^2 d}{b}+14 a c+\frac{b c^2}{d}\right )}{24 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{7/2}}{b^2 \sqrt{a+b x} (b c-a d)}-\frac{5 (b c-a d)^2 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{64 b^5 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right )}{96 b^4 d}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b*c - a*d)*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(64*b^5*d) - (5*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/
2))/(96*b^4*d) - ((14*a*c + (b*c^2)/d - (63*a^2*d)/b)*Sqrt[a + b*x]*(c + d*x)^(5
/2))/(24*b^2*(b*c - a*d)) - (2*a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*Sqrt[a + b*
x]) + (Sqrt[a + b*x]*(c + d*x)^(7/2))/(4*b^2*d) - (5*(b*c - a*d)^2*(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])])
/(64*b^(11/2)*d^(3/2))

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Rubi in Sympy [A]  time = 60.2702, size = 289, normalized size = 0.94 \[ \frac{2 a^{2} \left (c + d x\right )^{\frac{7}{2}}}{b^{2} \sqrt{a + b x} \left (a d - b c\right )} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{7}{2}}}{4 b^{2} d} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (63 a^{2} d^{2} - 14 a b c d - b^{2} c^{2}\right )}{24 b^{3} d \left (a d - b c\right )} + \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (63 a^{2} d^{2} - 14 a b c d - b^{2} c^{2}\right )}{96 b^{4} d} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (63 a^{2} d^{2} - 14 a b c d - b^{2} c^{2}\right )}{64 b^{5} d} + \frac{5 \left (a d - b c\right )^{2} \left (63 a^{2} d^{2} - 14 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 b^{\frac{11}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*(c + d*x)**(7/2)/(b**2*sqrt(a + b*x)*(a*d - b*c)) + sqrt(a + b*x)*(c + d*
x)**(7/2)/(4*b**2*d) - sqrt(a + b*x)*(c + d*x)**(5/2)*(63*a**2*d**2 - 14*a*b*c*d
- b**2*c**2)/(24*b**3*d*(a*d - b*c)) + 5*sqrt(a + b*x)*(c + d*x)**(3/2)*(63*a**
2*d**2 - 14*a*b*c*d - b**2*c**2)/(96*b**4*d) - 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*
d - b*c)*(63*a**2*d**2 - 14*a*b*c*d - b**2*c**2)/(64*b**5*d) + 5*(a*d - b*c)**2*
(63*a**2*d**2 - 14*a*b*c*d - b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqr
t(a + b*x)))/(64*b**(11/2)*d**(3/2))

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Mathematica [A]  time = 0.294724, size = 241, normalized size = 0.79 \[ \frac{\sqrt{c+d x} \left (-945 a^4 d^3+105 a^3 b d^2 (17 c-3 d x)+a^2 b^2 d \left (-839 c^2+637 c d x+126 d^2 x^2\right )+a b^3 \left (15 c^3-337 c^2 d x-244 c d^2 x^2-72 d^3 x^3\right )+b^4 x \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^5 d \sqrt{a+b x}}-\frac{5 (b c-a d)^2 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{11/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(-945*a^4*d^3 + 105*a^3*b*d^2*(17*c - 3*d*x) + a^2*b^2*d*(-839*c^
2 + 637*c*d*x + 126*d^2*x^2) + a*b^3*(15*c^3 - 337*c^2*d*x - 244*c*d^2*x^2 - 72*
d^3*x^3) + b^4*x*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 + 48*d^3*x^3)))/(192*b^5*
d*Sqrt[a + b*x]) - (5*(b*c - a*d)^2*(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]*Sqrt[c + d*x]])/(128*b^(11/2)*
d^(3/2))

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Maple [B]  time = 0.043, size = 961, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(d*x+c)^(1/2)*(96*x^4*b^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-144*x^3*
a*b^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+272*x^3*b^4*c*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))*x*a^4*b*d^4-2100*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*b^2*c*d^3+1350*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^3*c^2*d^2-180*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^4*c^3*
d-15*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^5*c^4+252*x^2*a^2*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-488*x^2*a*b^3
*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+236*x^2*b^4*c^2*d*((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^4-2100*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^3+1350*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*b^2*c^2*d^2-180*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*c^3*d-15*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*c^4-63
0*x*a^3*b*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1274*x*a^2*b^2*c*d^2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)-674*x*a*b^3*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
+30*x*b^4*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1890*a^4*d^3*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+3570*a^3*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1678*a^2
*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*a*b^3*c^3*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/d/(b*d)^(1/2)/(b*x+a)^(1/2)/b^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.951169, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{4} d^{3} x^{4} + 15 \, a b^{3} c^{3} - 839 \, a^{2} b^{2} c^{2} d + 1785 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 8 \,{\left (17 \, b^{4} c d^{2} - 9 \, a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (59 \, b^{4} c^{2} d - 122 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} +{\left (15 \, b^{4} c^{3} - 337 \, a b^{3} c^{2} d + 637 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (a b^{4} c^{4} + 12 \, a^{2} b^{3} c^{3} d - 90 \, a^{3} b^{2} c^{2} d^{2} + 140 \, a^{4} b c d^{3} - 63 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 12 \, a b^{4} c^{3} d - 90 \, a^{2} b^{3} c^{2} d^{2} + 140 \, a^{3} b^{2} c d^{3} - 63 \, a^{4} b d^{4}\right )} x\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 )}{768 \,{\left (b^{6} d x + a b^{5} d\right )} \sqrt{b d}}, \frac{2 \,{\left (48 \, b^{4} d^{3} x^{4} + 15 \, a b^{3} c^{3} - 839 \, a^{2} b^{2} c^{2} d + 1785 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 8 \,{\left (17 \, b^{4} c d^{2} - 9 \, a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (59 \, b^{4} c^{2} d - 122 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} +{\left (15 \, b^{4} c^{3} - 337 \, a b^{3} c^{2} d + 637 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left (a b^{4} c^{4} + 12 \, a^{2} b^{3} c^{3} d - 90 \, a^{3} b^{2} c^{2} d^{2} + 140 \, a^{4} b c d^{3} - 63 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 12 \, a b^{4} c^{3} d - 90 \, a^{2} b^{3} c^{2} d^{2} + 140 \, a^{3} b^{2} c d^{3} - 63 \, a^{4} b d^{4}\right )} x\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 )}{384 \,{\left (b^{6} d x + a b^{5} d\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*b^4*d^3*x^4 + 15*a*b^3*c^3 - 839*a^2*b^2*c^2*d + 1785*a^3*b*c*d^2
- 945*a^4*d^3 + 8*(17*b^4*c*d^2 - 9*a*b^3*d^3)*x^3 + 2*(59*b^4*c^2*d - 122*a*b^3
*c*d^2 + 63*a^2*b^2*d^3)*x^2 + (15*b^4*c^3 - 337*a*b^3*c^2*d + 637*a^2*b^2*c*d^2
- 315*a^3*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(a*b^4*c^4 + 12*
a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b*c*d^3 - 63*a^5*d^4 + (b^5*c^4 + 1
2*a*b^4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140*a^3*b^2*c*d^3 - 63*a^4*b*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^6*d*x
+ a*b^5*d)*sqrt(b*d)), 1/384*(2*(48*b^4*d^3*x^4 + 15*a*b^3*c^3 - 839*a^2*b^2*c^2
*d + 1785*a^3*b*c*d^2 - 945*a^4*d^3 + 8*(17*b^4*c*d^2 - 9*a*b^3*d^3)*x^3 + 2*(59
*b^4*c^2*d - 122*a*b^3*c*d^2 + 63*a^2*b^2*d^3)*x^2 + (15*b^4*c^3 - 337*a*b^3*c^2
*d + 637*a^2*b^2*c*d^2 - 315*a^3*b*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c
) - 15*(a*b^4*c^4 + 12*a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b*c*d^3 - 63
*a^5*d^4 + (b^5*c^4 + 12*a*b^4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140*a^3*b^2*c*d^3 -
63*a^4*b*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^6*d*x + a*b^5*d)*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**2*(d*x+c)**(5/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.638236, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x