Optimal. Leaf size=703 \[ -\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{e \left (a+b x+c x^2\right )^{7/4} \left (-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right )}{462 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{120 c^4}-\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{80 c^{9/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.866685, antiderivative size = 703, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {742, 779, 612, 623, 305, 220, 1196} \[ \frac{e \left (a+b x+c x^2\right )^{7/4} \left (-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right )}{462 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{120 c^4}-\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{80 c^{9/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 742
Rule 779
Rule 612
Rule 623
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/4} \, dx &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{2 \int (d+e x) \left (\frac{1}{4} \left (22 c d^2-7 b d e-8 a e^2\right )+\frac{15}{4} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4} \, dx}{11 c}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}+\frac{\left ((2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/4} \, dx}{24 c^3}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right )\right ) \int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx}{160 c^4}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{40 c^4 (b+2 c x)}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\left (\left (b^2-4 a c\right )^{3/2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{80 c^{9/2} (b+2 c x)}+\frac{\left (\left (b^2-4 a c\right )^{3/2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{2 \sqrt{c} x^2}{\sqrt{b^2-4 a c}}}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{80 c^{9/2} (b+2 c x)}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\sqrt{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{80 c^{9/2} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}\\ \end{align*}
Mathematica [C] time = 0.636513, size = 234, normalized size = 0.33 \[ \frac{77 (b+2 c x) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \left (8 c (a+x (b+c x))-3 \sqrt{2} \left (b^2-4 a c\right ) \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )\right )+160 c^2 e (a+x (b+c x))^2 \left (-2 c e (24 a e+121 b d+35 b e x)+55 b^2 e^2+4 c^2 d (78 d+35 e x)\right )+13440 c^4 e (d+e x)^2 (a+x (b+c x))^2}{73920 c^5 \sqrt [4]{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.04, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}{\left (e x + d\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]