### 3.345 $$\int (d+e x)^{7/2} (b x+c x^2)^2 \, dx$$

Optimal. Leaf size=147 $\frac{2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac{4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac{4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{17/2}}{17 e^5}$

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2
*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(15/2))/(15*e^5)
+ (2*c^2*(d + e*x)^(17/2))/(17*e^5)

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Rubi [A]  time = 0.0722376, antiderivative size = 147, 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} $\frac{2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac{4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac{4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac{2 c^2 (d+e x)^{17/2}}{17 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]

[Out]

(2*d^2*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*e^5) - (4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^5) + (2
*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(15/2))/(15*e^5)
+ (2*c^2*(d + e*x)^(17/2))/(17*e^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 (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac{d^2 (c d-b e)^2 (d+e x)^{7/2}}{e^4}+\frac{2 d (c d-b e) (-2 c d+b e) (d+e x)^{9/2}}{e^4}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{13/2}}{e^4}+\frac{c^2 (d+e x)^{15/2}}{e^4}\right ) \, dx\\ &=\frac{2 d^2 (c d-b e)^2 (d+e x)^{9/2}}{9 e^5}-\frac{4 d (c d-b e) (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{15/2}}{15 e^5}+\frac{2 c^2 (d+e x)^{17/2}}{17 e^5}\\ \end{align*}

Mathematica [A]  time = 0.0960983, size = 124, normalized size = 0.84 $\frac{2 (d+e x)^{9/2} \left (85 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+34 b c e \left (72 d^2 e x-16 d^3-198 d e^2 x^2+429 e^3 x^3\right )+c^2 \left (1584 d^2 e^2 x^2-576 d^3 e x+128 d^4-3432 d e^3 x^3+6435 e^4 x^4\right )\right )}{109395 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(9/2)*(85*b^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 34*b*c*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^
2 + 429*e^3*x^3) + c^2*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4)))/(109395*e^
5)

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Maple [A]  time = 0.048, size = 141, normalized size = 1. \begin{align*}{\frac{12870\,{c}^{2}{x}^{4}{e}^{4}+29172\,bc{e}^{4}{x}^{3}-6864\,{c}^{2}d{e}^{3}{x}^{3}+16830\,{b}^{2}{e}^{4}{x}^{2}-13464\,bcd{e}^{3}{x}^{2}+3168\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-6120\,{b}^{2}d{e}^{3}x+4896\,bc{d}^{2}{e}^{2}x-1152\,{c}^{2}{d}^{3}ex+1360\,{b}^{2}{d}^{2}{e}^{2}-1088\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{109395\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(c*x^2+b*x)^2,x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*c^2*e^4*x^4+14586*b*c*e^4*x^3-3432*c^2*d*e^3*x^3+8415*b^2*e^4*x^2-6732*b*c*d*e^3*
x^2+1584*c^2*d^2*e^2*x^2-3060*b^2*d*e^3*x+2448*b*c*d^2*e^2*x-576*c^2*d^3*e*x+680*b^2*d^2*e^2-544*b*c*d^3*e+128
*c^2*d^4)/e^5

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Maxima [A]  time = 1.09866, size = 188, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{2} - 14586 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 8415 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 19890 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 12155 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{109395 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

2/109395*(6435*(e*x + d)^(17/2)*c^2 - 14586*(2*c^2*d - b*c*e)*(e*x + d)^(15/2) + 8415*(6*c^2*d^2 - 6*b*c*d*e +
b^2*e^2)*(e*x + d)^(13/2) - 19890*(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2)*(e*x + d)^(11/2) + 12155*(c^2*d^4 - 2
*b*c*d^3*e + b^2*d^2*e^2)*(e*x + d)^(9/2))/e^5

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Fricas [B]  time = 1.9592, size = 662, normalized size = 4.5 \begin{align*} \frac{2 \,{\left (6435 \, c^{2} e^{8} x^{8} + 128 \, c^{2} d^{8} - 544 \, b c d^{7} e + 680 \, b^{2} d^{6} e^{2} + 858 \,{\left (26 \, c^{2} d e^{7} + 17 \, b c e^{8}\right )} x^{7} + 33 \,{\left (802 \, c^{2} d^{2} e^{6} + 1564 \, b c d e^{7} + 255 \, b^{2} e^{8}\right )} x^{6} + 36 \,{\left (303 \, c^{2} d^{3} e^{5} + 1751 \, b c d^{2} e^{6} + 850 \, b^{2} d e^{7}\right )} x^{5} + 5 \,{\left (7 \, c^{2} d^{4} e^{4} + 5440 \, b c d^{3} e^{5} + 7786 \, b^{2} d^{2} e^{6}\right )} x^{4} - 10 \,{\left (4 \, c^{2} d^{5} e^{3} - 17 \, b c d^{4} e^{4} - 1802 \, b^{2} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{6} e^{2} - 68 \, b c d^{5} e^{3} + 85 \, b^{2} d^{4} e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{7} e - 68 \, b c d^{6} e^{2} + 85 \, b^{2} d^{5} e^{3}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

2/109395*(6435*c^2*e^8*x^8 + 128*c^2*d^8 - 544*b*c*d^7*e + 680*b^2*d^6*e^2 + 858*(26*c^2*d*e^7 + 17*b*c*e^8)*x
^7 + 33*(802*c^2*d^2*e^6 + 1564*b*c*d*e^7 + 255*b^2*e^8)*x^6 + 36*(303*c^2*d^3*e^5 + 1751*b*c*d^2*e^6 + 850*b^
2*d*e^7)*x^5 + 5*(7*c^2*d^4*e^4 + 5440*b*c*d^3*e^5 + 7786*b^2*d^2*e^6)*x^4 - 10*(4*c^2*d^5*e^3 - 17*b*c*d^4*e^
4 - 1802*b^2*d^3*e^5)*x^3 + 3*(16*c^2*d^6*e^2 - 68*b*c*d^5*e^3 + 85*b^2*d^4*e^4)*x^2 - 4*(16*c^2*d^7*e - 68*b*
c*d^6*e^2 + 85*b^2*d^5*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 21.8537, size = 590, normalized size = 4.01 \begin{align*} \begin{cases} \frac{16 b^{2} d^{6} \sqrt{d + e x}}{1287 e^{3}} - \frac{8 b^{2} d^{5} x \sqrt{d + e x}}{1287 e^{2}} + \frac{2 b^{2} d^{4} x^{2} \sqrt{d + e x}}{429 e} + \frac{424 b^{2} d^{3} x^{3} \sqrt{d + e x}}{1287} + \frac{916 b^{2} d^{2} e x^{4} \sqrt{d + e x}}{1287} + \frac{80 b^{2} d e^{2} x^{5} \sqrt{d + e x}}{143} + \frac{2 b^{2} e^{3} x^{6} \sqrt{d + e x}}{13} - \frac{64 b c d^{7} \sqrt{d + e x}}{6435 e^{4}} + \frac{32 b c d^{6} x \sqrt{d + e x}}{6435 e^{3}} - \frac{8 b c d^{5} x^{2} \sqrt{d + e x}}{2145 e^{2}} + \frac{4 b c d^{4} x^{3} \sqrt{d + e x}}{1287 e} + \frac{640 b c d^{3} x^{4} \sqrt{d + e x}}{1287} + \frac{824 b c d^{2} e x^{5} \sqrt{d + e x}}{715} + \frac{184 b c d e^{2} x^{6} \sqrt{d + e x}}{195} + \frac{4 b c e^{3} x^{7} \sqrt{d + e x}}{15} + \frac{256 c^{2} d^{8} \sqrt{d + e x}}{109395 e^{5}} - \frac{128 c^{2} d^{7} x \sqrt{d + e x}}{109395 e^{4}} + \frac{32 c^{2} d^{6} x^{2} \sqrt{d + e x}}{36465 e^{3}} - \frac{16 c^{2} d^{5} x^{3} \sqrt{d + e x}}{21879 e^{2}} + \frac{14 c^{2} d^{4} x^{4} \sqrt{d + e x}}{21879 e} + \frac{2424 c^{2} d^{3} x^{5} \sqrt{d + e x}}{12155} + \frac{1604 c^{2} d^{2} e x^{6} \sqrt{d + e x}}{3315} + \frac{104 c^{2} d e^{2} x^{7} \sqrt{d + e x}}{255} + \frac{2 c^{2} e^{3} x^{8} \sqrt{d + e x}}{17} & \text{for}\: e \neq 0 \\d^{\frac{7}{2}} \left (\frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5}\right ) & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)*(c*x**2+b*x)**2,x)

[Out]

Piecewise((16*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*b**2*d**4*x**2
*sqrt(d + e*x)/(429*e) + 424*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 916*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*
b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*b*c*d**7*sqrt(d + e*x)/(6435*e**4)
+ 32*b*c*d**6*x*sqrt(d + e*x)/(6435*e**3) - 8*b*c*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 4*b*c*d**4*x**3*sqrt(
d + e*x)/(1287*e) + 640*b*c*d**3*x**4*sqrt(d + e*x)/1287 + 824*b*c*d**2*e*x**5*sqrt(d + e*x)/715 + 184*b*c*d*e
**2*x**6*sqrt(d + e*x)/195 + 4*b*c*e**3*x**7*sqrt(d + e*x)/15 + 256*c**2*d**8*sqrt(d + e*x)/(109395*e**5) - 12
8*c**2*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*c**2*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*c**2*d**5*x**3*s
qrt(d + e*x)/(21879*e**2) + 14*c**2*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*c**2*d**3*x**5*sqrt(d + e*x)/1215
5 + 1604*c**2*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*c**2*d*e**2*x**7*sqrt(d + e*x)/255 + 2*c**2*e**3*x**8*sqrt(
d + e*x)/17, Ne(e, 0)), (d**(7/2)*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), True))

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Giac [B]  time = 1.35583, size = 1237, normalized size = 8.41 \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)^(7/2)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/765765*(7293*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*d^3*e^(-2) + 4862*(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*c*d^3*e^(-3) +
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 + 115
5*(x*e + d)^(3/2)*d^4)*c^2*d^3*e^(-4) + 7293*(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^2*d^2*e^(-2) + 1326*(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*c*d^2*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)*c^2*d^2*e^(-4) + 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*d*e^(-2) + 510*(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*d*e^(-3) + 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)*c^2*d*e^(-4) + 85*(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*e^(-2) + 34*(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*e^(-3) + 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 + 425425*(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^2*e^(-4))
*e^(-1)