3.357 \(\int \frac{(b x+c x^2)^3}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=244 \[ \frac{2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac{6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}-\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 d^3 \sqrt{d+e x} (c d-b e)^3}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7} \]

[Out]

(2*d^3*(c*d - b*e)^3*Sqrt[d + e*x])/e^7 - (2*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(3/2))/e^7 + (6*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(5/2))/(5*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^(7/2))/(7*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(3*e^7) - (6*c^2
*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7)

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Rubi [A]  time = 0.101605, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {698} \[ \frac{2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac{6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}-\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 d^3 \sqrt{d+e x} (c d-b e)^3}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/Sqrt[d + e*x],x]

[Out]

(2*d^3*(c*d - b*e)^3*Sqrt[d + e*x])/e^7 - (2*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(3/2))/e^7 + (6*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(5/2))/(5*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^(7/2))/(7*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(3*e^7) - (6*c^2
*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7)

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 (b x+c x^2\right )^3}{\sqrt{d+e x}} \, dx &=\int \left (\frac{d^3 (c d-b e)^3}{e^6 \sqrt{d+e x}}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) \sqrt{d+e x}}{e^6}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{5/2}}{e^6}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{9/2}}{e^6}+\frac{c^3 (d+e x)^{11/2}}{e^6}\right ) \, dx\\ &=\frac{2 d^3 (c d-b e)^3 \sqrt{d+e x}}{e^7}-\frac{2 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{3/2}}{e^7}+\frac{6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac{2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{7/2}}{7 e^7}+\frac{2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{3 e^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7}\\ \end{align*}

Mathematica [A]  time = 0.135167, size = 206, normalized size = 0.84 \[ \frac{2 \sqrt{d+e x} \left (5005 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-2145 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+9009 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-4095 c^2 (d+e x)^5 (2 c d-b e)-15015 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+15015 d^3 (c d-b e)^3+1155 c^3 (d+e x)^6\right )}{15015 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(15015*d^3*(c*d - b*e)^3 - 15015*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 9009*d*(c*d - b*
e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 2145*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d +
e*x)^3 + 5005*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 4095*c^2*(2*c*d - b*e)*(d + e*x)^5 + 1155*c^3
*(d + e*x)^6))/(15015*e^7)

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Maple [A]  time = 0.053, size = 286, normalized size = 1.2 \begin{align*} -{\frac{-2310\,{c}^{3}{x}^{6}{e}^{6}-8190\,b{c}^{2}{e}^{6}{x}^{5}+2520\,{c}^{3}d{e}^{5}{x}^{5}-10010\,{b}^{2}c{e}^{6}{x}^{4}+9100\,b{c}^{2}d{e}^{5}{x}^{4}-2800\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-4290\,{b}^{3}{e}^{6}{x}^{3}+11440\,{b}^{2}cd{e}^{5}{x}^{3}-10400\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+3200\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+5148\,{b}^{3}d{e}^{5}{x}^{2}-13728\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+12480\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-6864\,{b}^{3}{d}^{2}{e}^{4}x+18304\,{b}^{2}c{d}^{3}{e}^{3}x-16640\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+13728\,{b}^{3}{d}^{3}{e}^{3}-36608\,{b}^{2}c{d}^{4}{e}^{2}+33280\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{15015\,{e}^{7}}\sqrt{ex+d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^(1/2),x)

[Out]

-2/15015*(-1155*c^3*e^6*x^6-4095*b*c^2*e^6*x^5+1260*c^3*d*e^5*x^5-5005*b^2*c*e^6*x^4+4550*b*c^2*d*e^5*x^4-1400
*c^3*d^2*e^4*x^4-2145*b^3*e^6*x^3+5720*b^2*c*d*e^5*x^3-5200*b*c^2*d^2*e^4*x^3+1600*c^3*d^3*e^3*x^3+2574*b^3*d*
e^5*x^2-6864*b^2*c*d^2*e^4*x^2+6240*b*c^2*d^3*e^3*x^2-1920*c^3*d^4*e^2*x^2-3432*b^3*d^2*e^4*x+9152*b^2*c*d^3*e
^3*x-8320*b*c^2*d^4*e^2*x+2560*c^3*d^5*e*x+6864*b^3*d^3*e^3-18304*b^2*c*d^4*e^2+16640*b*c^2*d^5*e-5120*c^3*d^6
)*(e*x+d)^(1/2)/e^7

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Maxima [A]  time = 1.14775, size = 389, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (\frac{429 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac{143 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac{65 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} - 385 \,{\left (e x + d\right )}^{\frac{9}{2}} d + 990 \,{\left (e x + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac{5 \,{\left (231 \,{\left (e x + d\right )}^{\frac{13}{2}} - 1638 \,{\left (e x + d\right )}^{\frac{11}{2}} d + 5005 \,{\left (e x + d\right )}^{\frac{9}{2}} d^{2} - 8580 \,{\left (e x + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{4} - 6006 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{5} + 3003 \, \sqrt{e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/15015*(429*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^3/e^
3 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*
sqrt(e*x + d)*d^4)*b^2*c/e^4 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 138
6*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b*c^2/e^5 + 5*(231*(e*x + d)^(13/2)
- 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6
006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3/e^6)/e

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Fricas [A]  time = 2.19685, size = 630, normalized size = 2.58 \begin{align*} \frac{2 \,{\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e + 18304 \, b^{2} c d^{4} e^{2} - 6864 \, b^{3} d^{3} e^{3} - 315 \,{\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \, b^{2} c e^{6}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \, b^{2} c d e^{5} - 429 \, b^{3} e^{6}\right )} x^{3} + 6 \,{\left (320 \, c^{3} d^{4} e^{2} - 1040 \, b c^{2} d^{3} e^{3} + 1144 \, b^{2} c d^{2} e^{4} - 429 \, b^{3} d e^{5}\right )} x^{2} - 8 \,{\left (320 \, c^{3} d^{5} e - 1040 \, b c^{2} d^{4} e^{2} + 1144 \, b^{2} c d^{3} e^{3} - 429 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/15015*(1155*c^3*e^6*x^6 + 5120*c^3*d^6 - 16640*b*c^2*d^5*e + 18304*b^2*c*d^4*e^2 - 6864*b^3*d^3*e^3 - 315*(4
*c^3*d*e^5 - 13*b*c^2*e^6)*x^5 + 35*(40*c^3*d^2*e^4 - 130*b*c^2*d*e^5 + 143*b^2*c*e^6)*x^4 - 5*(320*c^3*d^3*e^
3 - 1040*b*c^2*d^2*e^4 + 1144*b^2*c*d*e^5 - 429*b^3*e^6)*x^3 + 6*(320*c^3*d^4*e^2 - 1040*b*c^2*d^3*e^3 + 1144*
b^2*c*d^2*e^4 - 429*b^3*d*e^5)*x^2 - 8*(320*c^3*d^5*e - 1040*b*c^2*d^4*e^2 + 1144*b^2*c*d^3*e^3 - 429*b^3*d^2*
e^4)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 115.984, size = 745, normalized size = 3.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e
**3 + 2*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (
d + e*x)**(7/2)/7)/e**3 + 6*b**2*c*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*
d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 6*b**2*c*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d*
*3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 6*b*c**2
*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d
+ e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 + 6*b*c**2*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d
+ e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11
/2)/11)/e**5 + 2*c**3*d*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x
)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 + 2*c**3*(-d**7/sq
rt(d + e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/
2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6)/e, Ne(e, 0)), ((b**3*x
**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7)/sqrt(d), True))

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Giac [A]  time = 1.3159, size = 412, normalized size = 1.69 \begin{align*} \frac{2}{15015} \,{\left (429 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} b^{3} e^{\left (-3\right )} + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} b^{2} c e^{\left (-4\right )} + 65 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{x e + d} d^{5}\right )} b c^{2} e^{\left (-5\right )} + 5 \,{\left (231 \,{\left (x e + d\right )}^{\frac{13}{2}} - 1638 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 5005 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 8580 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 6006 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} + 3003 \, \sqrt{x e + d} d^{6}\right )} c^{3} e^{\left (-6\right )}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/15015*(429*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*b^3*e^
(-3) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 3
15*sqrt(x*e + d)*d^4)*b^2*c*e^(-4) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2
- 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b*c^2*e^(-5) + 5*(231*(x*e + d
)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2
)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^3*e^(-6))*e^(-1)