3.2 \(\int \frac{(c+d x+e x^2)^2}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=161 \[ -\frac{2 (a+b x)^{5/2} \left (-6 a^2 e^2+6 a b d e+b^2 \left (-\left (2 c e+d^2\right )\right )\right )}{5 b^5}+\frac{4 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )}{3 b^5}+\frac{2 \sqrt{a+b x} \left (a^2 e-a b d+b^2 c\right )^2}{b^5}+\frac{4 e (a+b x)^{7/2} (b d-2 a e)}{7 b^5}+\frac{2 e^2 (a+b x)^{9/2}}{9 b^5} \]

[Out]

(2*(b^2*c - a*b*d + a^2*e)^2*Sqrt[a + b*x])/b^5 + (4*(b*d - 2*a*e)*(b^2*c - a*b*d + a^2*e)*(a + b*x)^(3/2))/(3
*b^5) - (2*(6*a*b*d*e - 6*a^2*e^2 - b^2*(d^2 + 2*c*e))*(a + b*x)^(5/2))/(5*b^5) + (4*e*(b*d - 2*a*e)*(a + b*x)
^(7/2))/(7*b^5) + (2*e^2*(a + b*x)^(9/2))/(9*b^5)

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Rubi [A]  time = 0.105497, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {698} \[ -\frac{2 (a+b x)^{5/2} \left (-6 a^2 e^2+6 a b d e+b^2 \left (-\left (2 c e+d^2\right )\right )\right )}{5 b^5}+\frac{4 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )}{3 b^5}+\frac{2 \sqrt{a+b x} \left (a^2 e-a b d+b^2 c\right )^2}{b^5}+\frac{4 e (a+b x)^{7/2} (b d-2 a e)}{7 b^5}+\frac{2 e^2 (a+b x)^{9/2}}{9 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^2*c - a*b*d + a^2*e)^2*Sqrt[a + b*x])/b^5 + (4*(b*d - 2*a*e)*(b^2*c - a*b*d + a^2*e)*(a + b*x)^(3/2))/(3
*b^5) - (2*(6*a*b*d*e - 6*a^2*e^2 - b^2*(d^2 + 2*c*e))*(a + b*x)^(5/2))/(5*b^5) + (4*e*(b*d - 2*a*e)*(a + b*x)
^(7/2))/(7*b^5) + (2*e^2*(a + b*x)^(9/2))/(9*b^5)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (c+d x+e x^2\right )^2}{\sqrt{a+b x}} \, dx &=\int \left (\frac{\left (b^2 c-a b d+a^2 e\right )^2}{b^4 \sqrt{a+b x}}+\frac{2 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right ) \sqrt{a+b x}}{b^4}+\frac{\left (-6 a b d e+6 a^2 e^2+b^2 \left (d^2+2 c e\right )\right ) (a+b x)^{3/2}}{b^4}+\frac{2 e (b d-2 a e) (a+b x)^{5/2}}{b^4}+\frac{e^2 (a+b x)^{7/2}}{b^4}\right ) \, dx\\ &=\frac{2 \left (b^2 c-a b d+a^2 e\right )^2 \sqrt{a+b x}}{b^5}+\frac{4 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right ) (a+b x)^{3/2}}{3 b^5}-\frac{2 \left (6 a b d e-6 a^2 e^2-b^2 \left (d^2+2 c e\right )\right ) (a+b x)^{5/2}}{5 b^5}+\frac{4 e (b d-2 a e) (a+b x)^{7/2}}{7 b^5}+\frac{2 e^2 (a+b x)^{9/2}}{9 b^5}\\ \end{align*}

Mathematica [A]  time = 0.15182, size = 155, normalized size = 0.96 \[ \frac{2 \sqrt{a+b x} \left (24 a^2 b^2 \left (2 e \left (7 c+e x^2\right )+7 d^2+6 d e x\right )-32 a^3 b e (9 d+2 e x)+128 a^4 e^2-4 a b^3 \left (21 c (5 d+2 e x)+x \left (21 d^2+27 d e x+10 e^2 x^2\right )\right )+b^4 \left (315 c^2+42 c x (5 d+3 e x)+x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )\right )\right )}{315 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x]*(128*a^4*e^2 - 32*a^3*b*e*(9*d + 2*e*x) + 24*a^2*b^2*(7*d^2 + 6*d*e*x + 2*e*(7*c + e*x^2)) -
4*a*b^3*(21*c*(5*d + 2*e*x) + x*(21*d^2 + 27*d*e*x + 10*e^2*x^2)) + b^4*(315*c^2 + 42*c*x*(5*d + 3*e*x) + x^2*
(63*d^2 + 90*d*e*x + 35*e^2*x^2))))/(315*b^5)

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Maple [A]  time = 0.005, size = 194, normalized size = 1.2 \begin{align*}{\frac{70\,{e}^{2}{x}^{4}{b}^{4}-80\,a{b}^{3}{e}^{2}{x}^{3}+180\,{b}^{4}de{x}^{3}+96\,{a}^{2}{b}^{2}{e}^{2}{x}^{2}-216\,a{b}^{3}de{x}^{2}+252\,{b}^{4}ce{x}^{2}+126\,{b}^{4}{d}^{2}{x}^{2}-128\,{a}^{3}b{e}^{2}x+288\,{a}^{2}{b}^{2}dex-336\,a{b}^{3}cex-168\,a{b}^{3}{d}^{2}x+420\,{b}^{4}cdx+256\,{a}^{4}{e}^{2}-576\,{a}^{3}bde+672\,{a}^{2}{b}^{2}ce+336\,{a}^{2}{b}^{2}{d}^{2}-840\,a{b}^{3}cd+630\,{c}^{2}{b}^{4}}{315\,{b}^{5}}\sqrt{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(b*x+a)^(1/2)*(35*b^4*e^2*x^4-40*a*b^3*e^2*x^3+90*b^4*d*e*x^3+48*a^2*b^2*e^2*x^2-108*a*b^3*d*e*x^2+126*b
^4*c*e*x^2+63*b^4*d^2*x^2-64*a^3*b*e^2*x+144*a^2*b^2*d*e*x-168*a*b^3*c*e*x-84*a*b^3*d^2*x+210*b^4*c*d*x+128*a^
4*e^2-288*a^3*b*d*e+336*a^2*b^2*c*e+168*a^2*b^2*d^2-420*a*b^3*c*d+315*b^4*c^2)/b^5

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Maxima [A]  time = 0.931278, size = 320, normalized size = 1.99 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{b x + a} c^{2} + 42 \, c{\left (\frac{5 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )} d}{b} + \frac{{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} e}{b^{2}}\right )} + \frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac{18 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x + a} a^{3}\right )} d e}{b^{3}} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 180 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 378 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(b*x + a)*c^2 + 42*c*(5*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*d/b + (3*(b*x + a)^(5/2) - 10*(b*
x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b^2) + 21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a
)*a^2)*d^2/b^2 + 18*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)
*d*e/b^3 + (35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 3
15*sqrt(b*x + a)*a^4)*e^2/b^4)/b

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Fricas [A]  time = 1.28292, size = 433, normalized size = 2.69 \begin{align*} \frac{2 \,{\left (35 \, b^{4} e^{2} x^{4} + 315 \, b^{4} c^{2} - 420 \, a b^{3} c d + 168 \, a^{2} b^{2} d^{2} + 128 \, a^{4} e^{2} + 10 \,{\left (9 \, b^{4} d e - 4 \, a b^{3} e^{2}\right )} x^{3} + 3 \,{\left (21 \, b^{4} d^{2} + 16 \, a^{2} b^{2} e^{2} + 6 \,{\left (7 \, b^{4} c - 6 \, a b^{3} d\right )} e\right )} x^{2} + 48 \,{\left (7 \, a^{2} b^{2} c - 6 \, a^{3} b d\right )} e + 2 \,{\left (105 \, b^{4} c d - 42 \, a b^{3} d^{2} - 32 \, a^{3} b e^{2} - 12 \,{\left (7 \, a b^{3} c - 6 \, a^{2} b^{2} d\right )} e\right )} x\right )} \sqrt{b x + a}}{315 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*e^2*x^4 + 315*b^4*c^2 - 420*a*b^3*c*d + 168*a^2*b^2*d^2 + 128*a^4*e^2 + 10*(9*b^4*d*e - 4*a*b^3*
e^2)*x^3 + 3*(21*b^4*d^2 + 16*a^2*b^2*e^2 + 6*(7*b^4*c - 6*a*b^3*d)*e)*x^2 + 48*(7*a^2*b^2*c - 6*a^3*b*d)*e +
2*(105*b^4*c*d - 42*a*b^3*d^2 - 32*a^3*b*e^2 - 12*(7*a*b^3*c - 6*a^2*b^2*d)*e)*x)*sqrt(b*x + a)/b^5

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Sympy [A]  time = 50.6685, size = 644, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2*a*c**2/sqrt(a + b*x) + 4*a*c*d*(-a/sqrt(a + b*x) - sqrt(a + b*x))/b + 4*a*c*e*(a**2/sqrt(a + b*
x) + 2*a*sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b**2 + 2*a*d**2*(a**2/sqrt(a + b*x) + 2*a*sqrt(a + b*x) - (a + b*
x)**(3/2)/3)/b**2 + 4*a*d*e*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b*x)**(5/2
)/5)/b**3 + 2*a*e**2*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x)**(5/
2)/5 - (a + b*x)**(7/2)/7)/b**4 + 2*c**2*(-a/sqrt(a + b*x) - sqrt(a + b*x)) + 4*c*d*(a**2/sqrt(a + b*x) + 2*a*
sqrt(a + b*x) - (a + b*x)**(3/2)/3)/b + 4*c*e*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2)
- (a + b*x)**(5/2)/5)/b**2 + 2*d**2*(-a**3/sqrt(a + b*x) - 3*a**2*sqrt(a + b*x) + a*(a + b*x)**(3/2) - (a + b
*x)**(5/2)/5)/b**2 + 4*d*e*(a**4/sqrt(a + b*x) + 4*a**3*sqrt(a + b*x) - 2*a**2*(a + b*x)**(3/2) + 4*a*(a + b*x
)**(5/2)/5 - (a + b*x)**(7/2)/7)/b**3 + 2*e**2*(-a**5/sqrt(a + b*x) - 5*a**4*sqrt(a + b*x) + 10*a**3*(a + b*x)
**(3/2)/3 - 2*a**2*(a + b*x)**(5/2) + 5*a*(a + b*x)**(7/2)/7 - (a + b*x)**(9/2)/9)/b**4)/b, Ne(b, 0)), ((c**2*
x + c*d*x**2 + d*e*x**4/2 + e**2*x**5/5 + x**3*(2*c*e + d**2)/3)/sqrt(a), True))

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Giac [A]  time = 1.1067, size = 320, normalized size = 1.99 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{b x + a} c^{2} + \frac{210 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )} c d}{b} + \frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} d^{2}}{b^{2}} + \frac{42 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x + a} a^{2}\right )} c e}{b^{2}} + \frac{18 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x + a} a^{3}\right )} d e}{b^{3}} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 180 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 378 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{b x + a} a^{4}\right )} e^{2}}{b^{4}}\right )}}{315 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)^2/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/315*(315*sqrt(b*x + a)*c^2 + 210*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*c*d/b + 21*(3*(b*x + a)^(5/2) - 10*(b
*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*d^2/b^2 + 42*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x +
a)*a^2)*c*e/b^2 + 18*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^
3)*d*e/b^3 + (35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 +
315*sqrt(b*x + a)*a^4)*e^2/b^4)/b