3.355 \(\int (d+e x)^{3/2} (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=248 \[ \frac{6 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{11 e^7}+\frac{2 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}-\frac{6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac{2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7} \]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(5/2))/(5*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) +
(2*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(3*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 1
0*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13
*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(5*e^7) + (2*c^3*(d + e*x)^(17/2))/(17*e^7)

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Rubi [A]  time = 0.105789, antiderivative size = 248, 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} \[ \frac{6 c (d+e x)^{13/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{11 e^7}+\frac{2 d (d+e x)^{9/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}-\frac{6 d^2 (d+e x)^{7/2} (c d-b e)^2 (2 c d-b e)}{7 e^7}+\frac{2 d^3 (d+e x)^{5/2} (c d-b e)^3}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(5/2))/(5*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) +
(2*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(9/2))/(3*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 - 1
0*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13
*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(5*e^7) + (2*c^3*(d + e*x)^(17/2))/(17*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 (d+e x)^{3/2} \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac{d^3 (c d-b e)^3 (d+e x)^{3/2}}{e^6}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{5/2}}{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)^{7/2}}{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)^{9/2}}{e^6}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{13/2}}{e^6}+\frac{c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac{2 d^3 (c d-b e)^3 (d+e x)^{5/2}}{5 e^7}-\frac{6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^7}+\frac{2 d (c d-b e) \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{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)^{11/2}}{11 e^7}+\frac{6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac{2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}\\ \end{align*}

Mathematica [A]  time = 0.161388, size = 206, normalized size = 0.83 \[ \frac{2 (d+e x)^{5/2} \left (58905 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-23205 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+85085 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-51051 c^2 (d+e x)^5 (2 c d-b e)-109395 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+51051 d^3 (c d-b e)^3+15015 c^3 (d+e x)^6\right )}{255255 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(51051*d^3*(c*d - b*e)^3 - 109395*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 85085*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 23205*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)
*(d + e*x)^3 + 58905*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 51051*c^2*(2*c*d - b*e)*(d + e*x)^5 + 1
5015*c^3*(d + e*x)^6))/(255255*e^7)

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Maple [A]  time = 0.049, size = 286, normalized size = 1.2 \begin{align*} -{\frac{-30030\,{c}^{3}{x}^{6}{e}^{6}-102102\,b{c}^{2}{e}^{6}{x}^{5}+24024\,{c}^{3}d{e}^{5}{x}^{5}-117810\,{b}^{2}c{e}^{6}{x}^{4}+78540\,b{c}^{2}d{e}^{5}{x}^{4}-18480\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-46410\,{b}^{3}{e}^{6}{x}^{3}+85680\,{b}^{2}cd{e}^{5}{x}^{3}-57120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+13440\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+30940\,{b}^{3}d{e}^{5}{x}^{2}-57120\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+38080\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-8960\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-17680\,{b}^{3}{d}^{2}{e}^{4}x+32640\,{b}^{2}c{d}^{3}{e}^{3}x-21760\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+7072\,{b}^{3}{d}^{3}{e}^{3}-13056\,{b}^{2}c{d}^{4}{e}^{2}+8704\,b{c}^{2}{d}^{5}e-2048\,{c}^{3}{d}^{6}}{255255\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/255255*(e*x+d)^(5/2)*(-15015*c^3*e^6*x^6-51051*b*c^2*e^6*x^5+12012*c^3*d*e^5*x^5-58905*b^2*c*e^6*x^4+39270*
b*c^2*d*e^5*x^4-9240*c^3*d^2*e^4*x^4-23205*b^3*e^6*x^3+42840*b^2*c*d*e^5*x^3-28560*b*c^2*d^2*e^4*x^3+6720*c^3*
d^3*e^3*x^3+15470*b^3*d*e^5*x^2-28560*b^2*c*d^2*e^4*x^2+19040*b*c^2*d^3*e^3*x^2-4480*c^3*d^4*e^2*x^2-8840*b^3*
d^2*e^4*x+16320*b^2*c*d^3*e^3*x-10880*b*c^2*d^4*e^2*x+2560*c^3*d^5*e*x+3536*b^3*d^3*e^3-6528*b^2*c*d^4*e^2+435
2*b*c^2*d^5*e-1024*c^3*d^6)/e^7

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Maxima [A]  time = 1.14933, size = 366, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (15015 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} - 51051 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 58905 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 23205 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 109395 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 51051 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{255255 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(15015*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d)^(15/2) + 58905*(5*c^3*d^2 - 5*b*c^2
*d*e + b^2*c*e^2)*(e*x + d)^(13/2) - 23205*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x + d)^
(11/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(9/2) - 109395*(2*c^3*d^5
- 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)^(7/2) + 51051*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^
4*e^2 - b^3*d^3*e^3)*(e*x + d)^(5/2))/e^7

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Fricas [A]  time = 2.03218, size = 841, normalized size = 3.39 \begin{align*} \frac{2 \,{\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e + 6528 \, b^{2} c d^{6} e^{2} - 3536 \, b^{3} d^{5} e^{3} + 3003 \,{\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \,{\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \, b^{2} c e^{8}\right )} x^{6} - 21 \,{\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \, b^{2} c d e^{7} - 1105 \, b^{3} e^{8}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \, b^{2} c d^{2} e^{6} + 884 \, b^{3} d e^{7}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} + 408 \, b^{2} c d^{3} e^{5} - 221 \, b^{3} d^{2} e^{6}\right )} x^{3} + 6 \,{\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} + 408 \, b^{2} c d^{4} e^{4} - 221 \, b^{3} d^{3} e^{5}\right )} x^{2} - 8 \,{\left (64 \, c^{3} d^{7} e - 272 \, b c^{2} d^{6} e^{2} + 408 \, b^{2} c d^{5} e^{3} - 221 \, b^{3} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}{255255 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(15015*c^3*e^8*x^8 + 1024*c^3*d^8 - 4352*b*c^2*d^7*e + 6528*b^2*c*d^6*e^2 - 3536*b^3*d^5*e^3 + 3003*(
6*c^3*d*e^7 + 17*b*c^2*e^8)*x^7 + 231*(c^3*d^2*e^6 + 272*b*c^2*d*e^7 + 255*b^2*c*e^8)*x^6 - 21*(12*c^3*d^3*e^5
- 51*b*c^2*d^2*e^6 - 3570*b^2*c*d*e^7 - 1105*b^3*e^8)*x^5 + 35*(8*c^3*d^4*e^4 - 34*b*c^2*d^3*e^5 + 51*b^2*c*d
^2*e^6 + 884*b^3*d*e^7)*x^4 - 5*(64*c^3*d^5*e^3 - 272*b*c^2*d^4*e^4 + 408*b^2*c*d^3*e^5 - 221*b^3*d^2*e^6)*x^3
+ 6*(64*c^3*d^6*e^2 - 272*b*c^2*d^5*e^3 + 408*b^2*c*d^4*e^4 - 221*b^3*d^3*e^5)*x^2 - 8*(64*c^3*d^7*e - 272*b*
c^2*d^6*e^2 + 408*b^2*c*d^5*e^3 - 221*b^3*d^4*e^4)*x)*sqrt(e*x + d)/e^7

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Sympy [B]  time = 34.4452, size = 738, normalized size = 2.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**3*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/
e**4 + 2*b**3*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)
**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 6*b**2*c*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d
**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 6*b**2*c*(-d**5*(d + e*x)**(3/2
)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/
11 + (d + e*x)**(13/2)/13)/e**5 + 6*b*c**2*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d +
e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 6*b*c**2*
(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9
+ 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 2*c**3*d*(d**6*(d +
e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*
(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 2*c**3*(-d**7*(d + e*x)**(3/2)/
3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/
2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7

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Giac [B]  time = 1.35408, size = 892, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(2431*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3
)*b^3*d*e^(-3) + 663*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d
)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^2*c*d*e^(-4) + 255*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d +
10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*
b*c^2*d*e^(-5) + 17*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x
*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c^3*d*e
^(-6) + 221*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d
^3 + 1155*(x*e + d)^(3/2)*d^4)*b^3*e^(-3) + 255*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e +
d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^2*c*e^(-4)
+ 51*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d
^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*b*c^2*e^(-5) + 7*(6435
*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425
425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)
*c^3*e^(-6))*e^(-1)